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Real Economic Shocks and Sovereign Credit Risk∗Forthcoming, Journal of Financial and Quantitative Analysis
Patrick Augustin† Romeo Tedongap‡
McGill University Stockholm School of Economics
Abstract
We provide new empirical evidence that U.S. expected growth and consumption
volatility are closely related to the strong co-movement in sovereign spreads. We ra-
tionalize these findings in an equilibrium model with recursive utility for CDS spreads.
The framework nests a reduced-form default process with country-specific sensitivity to
expected growth and macroeconomic uncertainty. Exploiting the high-frequency infor-
mation in the CDS term structure across 38 countries, we estimate the model and find
parameters consistent with preference for early resolution of uncertainty. Our results
confirm the existence of time-varying risk premia in sovereign spreads as compensation
for exposure to common U.S. macroeconomic risk.
Keywords: CDS, Generalized Disappointment Aversion, Sovereign Risk, Term StructureJEL Classification: C5, E44, F30, G12, G15
∗The authors would like to thank the Managing Editor Stephen J. Brown and an anonymous referee fortheir valuable inputs to this draft of the paper. We would also like to thank David Backus, Steven Baker,Nina Boyarchenko, Tobias Broer, Anna Cieslak, Mike Chernov, Magnus Dahlquist, Rene Garcia, JørgenHaug, Kris Jacobs, David Lando, Belen Nieto, Stijn Van Nieuwerburgh, Paoloa Zerilli, Stanley Zin andseminar participants at the Stockholm School of Economics, the London School of Economics AIRC, theIRMC Amsterdam, the Nordic Finance Network, the EDHEC Business School, the Bank of Spain/Bankof Canada Workshop on Advances in Fixed Income Modeling, the New York Fed, the NYU Stern Macro-Finance Workshop, the Queen Mary University London, the 2012 Meeting of the Econometric Society, the2012 Financial Econometrics Conference in Toulouse, the European Economic Association 2012 Malaga,the 3rd International Conference of the Financial Engineering and Banking Society in Paris and the 2013Northern Finance Association for helpful comments. The present project has been supported by the NationalResearch Fund Luxembourg, the Nasdaq-OMX Nordic Foundation, the Jan Wallander and Tom HedeliusFoundation, as well as the Bank Research Institute in Stockholm.†McGill University - Desautels Faculty of Management, 1001 Sherbrooke St. West, Montreal, Quebec
H3A 1G5, Canada. Email: Patrick.Augustin@mcgill.ca.‡Swedish House of Finance and Stockholm School of Economics, Drottninggatan 98, SE-111 60 Stockholm,
Sweden. Email: Romeo.Tedongap@hhs.se.
I Introduction
Sovereign credit spreads co-move significantly over time. In light of this fact, there is sub-
stantial evidence that global factors have strong explanatory power for sovereign credit risk
(in particular at higher frequencies), above and beyond that of country-specific fundamen-
tals. The recent literature has emphasized the United States (U.S.) financial channel as
a global source of risk (Pan and Singleton (2008), Longstaff, Pan, Pedersen, and Singleton
(2010)). Whether the ultimate source is financial or macroeconomic in nature is nevertheless
a debate (Ang and Longstaff (2013)).1 We emphasize the macroeconomic channel through
new empirical evidence on a tight relationship of expected U.S. consumption growth and
consumption volatility with sovereign credit risk. We rationalize this evidence by embedding
a reduced-form default process driven by expected growth and consumption volatility into
an equilibrium model for credit default swap (CDS) spreads. The model’s ability to match
several dimensions of the sovereign CDS market corroborates the existence of time-varying
risk premia as compensation for exposure to (common) U.S. macroeconomic risk.
We empirically document a strong association between sovereign and global macroeco-
nomic risk, which cannot be accounted for by global financial risk. To demonstrate our re-
sults, we exploit the homogenous contract specification and daily variation in CDS spreads.2
Our sample contains information on the full term structure for a geographically dispersed
panel of 38 countries. The motivation for our analysis is based on the fact that sovereign
defaults tend to cluster at business cycle frequencies (Reinhart and Rogoff (2008)). First, we
show that the average level of spreads has a surprisingly high correlation of respectively -65%
and 85% with expected growth rates and economic uncertainty in the U.S. Second, the same
two risk factors in turn can account for approximately 75% of the variation in the first two
common factors embedded in the term structure across all 38 countries. We believe the two
1Ang and Longstaff (2013) focus on systemic sovereign risk.
2CDS spreads are better comparable across countries than sovereign bonds, as they are not plagued by
differences in covenants, issuer country legislation or declining spread maturities.
1
factors to be a level and a slope factor. Importantly, this association is robust against a bat-
tery of financial market variables, such as the CBOE S&P 500 volatility index, the variance
risk premium, the U.S. excess equity return, the price-earnings ratio as well as the high-yield
and investment-grade bond spreads. While some of these factors have individually statistical
explanatory power for the level factor, none of them accounts for the variation in the slope
factor. Third, we document that our results are qualitatively and quantitatively maintained
for country-specific regressions, where we test for the level and slope of the CDS term struc-
ture as regressands. The empirical evidence survives several robustness tests, among which
one suggests that expected growth and consumption volatility contemporaneously predict
the Reinhart and Rogoff sovereign distress indicator over the past 50 years.
These new empirical findings suggest that consumption risks are priced factors in the
term structure of sovereign CDS spreads and that time-variation in expected growth and
consumption volatility are strongly related to the significant co-movement in spreads. We
rationalize this result through a general equilibrium framework for CDS spreads that embeds
a reduced-form default process driven by aggregate consumption risk factors. Our rational
investor is risk averse and exhibits Epstein and Zin (1989) (EZ) recursive preferences. We also
extend the model to accommodate generalized disappointment averse (GDA) preferences as
in Routledge and Zin (2010). The conditional mean and volatility of aggregate consumption
growth, the two state variables of our endowment economy, drive the default process, while
countries differ cross-sectionally through their sensitivity to the systematic risk factors. As
the model yields closed-form solutions for CDS spreads and their moments, we are able
to estimate the structural parameters of the model. In addition, country spreads can be
interpreted based on preferences and exposures to macroeconomic risk factors.
To investigate the asset pricing implications, we exploit the high-frequency information
in CDS spreads from May 2003 through August 2010 and estimate both preference param-
eters and the cross-sectional default sensitivities to expected growth rates and consumption
volatility. We quantitatively match the unconditional mean, volatility, skewness and persis-
2
tence of the term structure, the decreasing pattern of the kurtosis with asset maturity as well
as historically observed cumulative default probabilities. A two-factor model for the default
process is necessary to account for these stylized facts, as a specification based only on ex-
pected consumption growth consistently produces a term structure, which is too steep, while
it is too low if macroeconomic uncertainty is the only source of risk. Adding country-specific
shocks to the model only marginally improves the fit. In addition to average upward sloping
term structures, the model qualitatively matches the conditional slope reversals for distressed
sovereign borrowers in states of bad macroeconomic fundamentals. Both the EZ and GDA
scenarios produce comparable unconditional results. However, a model with downside risk
aversion endogenously generates relatively higher risk aversion in states of bad macroeco-
nomic fundamentals, which yields a stronger term structure inversion in the worst economic
state that is quantitatively consistent with empirically observed magnitudes. A likelihood
ratio test favors asymmetric preferences over symmetric gambles. State-dependent prices
in bad states further reflect the tail risk embedded in CDS spreads. Moreover, the model
produces ratios of risk-neutral to physical default probabilities consistent with the literature.
Regarding the preference parameters in the EZ (GDA) case, we estimate an elasticity
of intertemporal substitution of 1.58 (1.49) and a coefficient of relative risk aversion of 8.27
(4.91), consistent with preference for early resolution of uncertainty. In the GDA case, we
estimate a disappointment aversion parameter of 0.25 and a disappointment threshold at 85%
of the certainty equivalent. We continue to match first and second moments of the stock
market and risk-free rate. We thus provide a joint framework for pricing equity and fixed
income derivatives, consistent with evidence of information flow between both markets.3
The results suggest that a simple two-factor model including expected growth and macroe-
conomic uncertainty is sufficient to explain a large fraction of the common time-variation
in the global sovereign CDS market. While the previous literature has emphasized the im-
3Two recent papers addressing the equity and cash credit spread puzzle in a unified consumption-based
framework are Bhamra, Kuehn, and Strebulaev (2010) and Chen, Collin-Dufresne, and Goldstein (2009).
3
portance of U.S. financial market variables for the common variation in spreads (Pan and
Singleton (2008), Longstaff, Pan, Pedersen, and Singleton (2010), Ang and Longstaff (2013)),
we focus on a different channel: shocks to the expected level and volatility of U.S. consump-
tion growth. We explain our findings in a tractable two-state model.4 Surprisingly, Pan
and Singleton (2008) address the importance of consumption risk in their discussion, yet
don’t include it directly in their analysis. Other evidence on the role of global factors in
explaining the co-movement in sovereign spreads points to links through U.S. interest rates
(Uribe and Yue (2006)), investors’ risk appetite (Baek, Bandopadhyaya and Du (2005), Re-
molona, Scatigna, and Wu (2008), Gonzalez-Rozada and Yeyati (2008)), investor sentiment
(Weigel and Gemmill (2006)), aggregate credit risk (Geyer, Kossmeier, and Pichler (2004))
or contagion (Benzoni, Collin-Dufresne, Goldstein, and Helwege (2012)).
Benzoni, Collin-Dufresne, Goldstein, and Helwege (2012) develop an equilibrium model
for CDSs with fragile beliefs as in Hansen (2007) and Hansen and Sargent (2010). The
co-movement in spreads arises through a hidden contagion factor that affects the posterior
default distribution of all countries after a shock to one individual country. They focus on
fitting the dynamics of the level of spreads, while we show that a simple two-factor model
matches higher order moments of the term structure. Our model is conceptually also close
to Borri and Verdelhan (2009), who apply the Campbell and Cochrane (1999) external habit
framework to show that emerging market bonds reflect a risk premium for exposure to the
U.S. market returns. Their objective is closer to the extensive literature on the sovereign
incentives to default and they fail to match the level of spreads. We focus on the pricing
of spreads and match the moments of the term structure up to the fourth order closely.
Moreover, we derive analytic solutions for non-linear CDS payoffs, allowing us to estimate
the default and preference parameters in the model.
We believe our results to be useful in better understanding the determinants of the
4The consumption correlation puzzle of Backus, Kehoe, and Kydland (1992) illustrates that consumption
growth is only weakly correlated across countries.
4
term structure of sovereign credit risk. Improving this understanding is warranted in light
of the developments in the emerging markets in the nineties, and particularly the recent
sovereign debt crisis in Europe with six international bailouts in approximately three years.
Understanding the relationship between higher frequency variation in sovereign spreads and
global risk factors is also relevant for the diversification benefits of international investors
and the efficiency of government intervention aimed at reducing sovereign borrowing costs.
The rest of the paper proceeds as follows. Section 2 provides new empirical evidence on
a tight link between sovereign credit and aggregate macroeconomic risk. Section 3 presents
the general equilibrium framework for CDS spreads. The model estimation is explained in
section 4, followed by a discussion of the results in section 5. We conclude in section 6.
II U.S. Consumption Risk and Sovereign CDS Spreads
Reinhart and Rogoff (2008) document over a 200-year period that sovereign defaults tend to
cluster at business-cycle frequencies. This motivates us to investigate more closely the link
between sovereign credit risk, for which we exploit the daily information on sovereign distress
embedded in CDS spreads, and global macroeconomic risk, proxied through the conditional
mean and volatility of aggregate U.S. consumption growth. The next paragraph describes
our CDS dataset, followed by a deeper study of its link with global macroeconomic risk.
A CDS Data and Summary Statistics
A CDS is a fixed income derivative, which allows a protection buyer to purchase insurance
from the protection seller against a contingent credit event on a reference entity by paying
a premium, referred to as the CDS spread.5 CDSs are particularly useful to study sovereign
credit risk across countries as the contract specification is identical for each sovereign. Pub-
lic bonds are usually characterized through differences in covenants and are often issued in
5Pan and Singleton (2008) describe the structure and institutional details of the sovereign CDS market.
5
different legal jurisdictions. In addition, CDSs are constant-maturity products with price in-
formation available at daily frequencies. This makes them suitable for studying the common
variation in sovereign spreads, which is particularly pronounced at higher frequencies.
Our data set consists of daily mid composite USD denominated CDS prices for 38
sovereign countries from Markit over the sample period May 9th, 2003 until August 19th,
2010, and covers prices for the full term structure, including 1, 2, 3, 5, 7 and 10-year con-
tracts.6,7 All contracts contain the full restructuring clause. The sample is representative
of the global sovereign CDS market as it spans 4 major geographical regions and 17 rating
categories. The list of 38 countries is provided in Table 1. Keeping track of rating changes
over time, there are on average 4 countries in rating category AAA, 6 in AA, 9 in A, 11 in
BBB, 6 in BB and 2 in rating group B.8
Our working data set thus contains 1900 observations for 38 countries and 6 maturities,
amounting to a total of 433,200 observations. We aggregate countries by rating categories
and present summary statistics in Table 2.9 The dataset exhibits a considerable amount of
heterogeneity both across time and across countries. For instance, the mean 5-year spread for
AAA rated countries is 22 basis points and 558 basis points for B rated sovereigns. The mean
term structure is always upward sloping, increasing monotonically with the deterioration of
credit quality from 11 (AAA) to 177 basis points (B). The sample features great time-series
variation, with standard deviations for the 5-year series ranging from 31 (AAA) to 287
basis points (B). All time series are highly persistent, with an average daily autocorrelation
coefficient for 5-year spreads as high as 0.9965.
6We fill gaps with CDS information from CMA Datastream and interpolate if no information is available.
7While trading in corporate CDS is highly concentrated around the 5-year maturity, Pan and Singleton
(2008) document that trading volume in sovereign CDS is more balanced across maturities.
8Venezuela is excluded for the 31 first days of the sample period when it was rated CCC+.
9We map ratings into a numerical scheme ranging from 1 (AAA) to 21 (C). The daily rating-specific
spread is the equally weighted mean spread of countries in a rating group.
6
B The Role of Macroeconomic Risk
As a first exercise to investigate the relationship between sovereign and global macroeconomic
risk, we test how strongly U.S. consumption risk factors are related to the common factors
in the sovereign CDS term structure. A principal component analysis (PCA) on the level
of spreads yields that the first two factors account for approximately 91% of the common
variation. This is rather strong, given the wide spectrum of contract maturities and reference
entities we consider. The magnitude is consistent with Pan and Singleton (2008), who
find that the first principal component explains on average 96% of the (daily) common
variation for Korea, Turkey and Mexico, and Longstaff, Pan, Pedersen, and Singleton (2010),
who document an average explained (monthly) variation by the first factor of 64% for 26
countries, increasing to 75% during the financial crisis. Table 3 illustrates the proportion
of the variance explained by the first six principal components. The magnitudes remain
similar for subsamples of the term structure. The importance of the first factor decreases
nevertheless relative to the second factor as we move towards longer maturities.
To summarize the information from the PCA, we average the country factor loadings
across maturities. Figure 1 shows that the average factor loadings on the first principal
component have roughly equal magnitudes, whereas those on the second factor increase
monotonically with maturity. We therefore interpret the first and second factors as a level
and slope factor in the CDS term structure.
If U.S. consumption risk is priced in the sovereign CDS market, then it ought to be linked
to the factors extracted from this PCA. As insurance premia on contingent future default
events may be linked to expectations about future growth rates, we investigate both channels
of time-varying expected consumption growth and macroeconomic uncertainty. More specif-
ically, we use monthly real per capita consumption data from January 1959 until August
7
2010, available at the Federal Reserve Bank of St.Louis, to estimate the system of equations
gt+1 = xt + σtεg,t+1(1)
xt+1 = (1− φx)µx + φxxt + νxσtεx,t+1
σ2t+1 = (1− φσ)µσ + φσσ
2t + νσεσ,t+1,
where gt, xt and σt reflect the dynamics for, respectively, aggregate consumption growth,
its conditional mean and volatility, and all error terms are standard normal. In addition
to the short-run consumption shocks εg,t+1, consumption growth is fed with long-run shocks
εx,t+1, whose persistence is defined through the parameter φx. The unconditional mean
growth-rate is given by µx and νx denotes the sensitivity of expected growth to long-run
shocks. The parameters φσ and νσ define the persistence of and sensitivity to shocks εσ,t+1
to macroeconomic uncertainty, which is fluctuating around its long-run mean µσ. We define
macroeconomic uncertainty as a GARCH-like stochastic volatility process that has been used
in Heston and Nandi (2000), henceforth HN, to price options. Such dynamics have recently
been applied in Tedongap (2014) to illustrate the relationship between consumption volatility
and the cross-section of stock returns.10 We note that the volatility shocks are directly related
to the realized consumption shocks, but this does not drive our empirical results, as we later
discuss. The results are also robust to other volatility filters, which we describe in the
internet appendix because of space limitations.
We adopt a Kalman Filter as in Hamilton (1994) to filter a (unsmoothed) time series
for the conditional expected consumption growth xt|t, where xt|t denotes the expectation
conditional on observed consumption growth up to time t, and the conditional consumption
volatility σt. The parameter estimates reported in Panel A of Table 4 are comparable to
those used in standard calibration exercises of long-run risk frameworks. As the highest
10Note that in contrast to Heston and Nandi (2000), we restrict the leverage parameter to zero. In
addition, the volatility process is strictly positive as long as (1− φσ)µσ − νσ/√
2 ≥ 0. The residuals are
normalized to have zero mean and unit variance, i.e. εσ,t+1 =(ε2g,t+1 − 1
)/√
2.
8
available frequency for consumption information is monthly, we regress monthly averages of
the first two factor scores Fi,t onto conditional expected growth and consumption volatility
Fi,t = a0,i + a1,i × xt|t + a2,i × σt + εt,(2)
where i = 1, 2 and t is the month index. Results based on 88 monthly observations with
block-bootstrapped standard errors are reported in columns (1) to (2) of Table 5.
Both coefficients on the explanatory variables are statistically significant at the 1% sig-
nificance level for regressions (1) and (2). In addition, the adjusted R2 from the first two
regressions is 75% and 74%, respectively. On a stand-alone basis, U.S. consumption risk
seems strongly associated with the first two common components in the CDS term struc-
ture, which themselves explain on average almost 91% of the time-variation in sovereign
spreads. We will discuss the signs of the individual regression coefficients in relation to the
model-implied results on the assumption that the first two principal components reflect a
level and a slope factor. An exact interpretation of the economic magnitudes of the latent
factor loadings in relation to the common factors is, however, more difficult.
Our results support the view that expected U.S. consumption growth and volatility are
two risk factors in the term structure of sovereign credit spreads.11 Macroeconomic shocks
channel through to the level and the slope of the term structure. A back-of-the-envelope
calculation yields that shocks to expected U.S. consumption growth and volatility manage to
explain roughly 68.25% of the common variation in sovereign CDS premia.12 Our conclusion
is visually emphasized by plotting the filtered series of conditional expected consumption
growth and consumption volatility against the average monthly 5-year CDS spread of all 38
countries in Figure 2. There is a strong negative correlation (-65%) between the average
11We acknowledge an error-in-variables problem in the factor regressions, but a simultaneous estimation
of the regression coefficients and the risk factors is unlikely to undo our strong results.
12This back-of-the-envelope calculation is based on the filtered consumption series, which explains 75%
of the first two principal components, which themselves account for 91% of the variation in spreads.
9
sovereign spread and conditional expected consumption growth. Moreover, the conditional
consumption volatility tracks the mean 5-year CDS spread closely with a staggering corre-
lation of 85%.
In order to obtain further empirical support for the relationship between sovereign credit
risk and both expected growth and consumption volatility over a longer horizon than the CDS
data availability, we exploit the crisis tally indicator of Reinhart and Rogoff (2008).13 The
crisis tally is a count indicator accounting for currency crises, inflation crises, stock market
crashes, domestic and external sovereign debt crises, and banking crises. It is available for
the entire period where we have monthly consumption growth data and for 29 countries in
our sample. Therefore we fit a probit model to test whether the two US macroeconomic risk
factors explain the outcome of the crisis tally indicator.14 More specifically, we fit the model
Pr (Crisis = 1) = Φ(a0 + a1xt|t + a2σt
),(3)
and report the results using block-bootstrapped standard errors in column (15) of Table
5. Both coefficients for the conditional mean and volatility of consumption growth are
significant at the 1% level. Evaluating all explanatory variables at their median value, the
coefficients of -0.22 and 0.45 correspond to marginal effects of -0.09 and 0.18, respectively.
This implies that, if all values are at their median, a one percentage point increase in expected
consumption growth decreases the likelihood of a crisis by 0.09, ceteris paribus. Likewise,
a one percentage point rise in consumption volatility increases the probability of a crisis by
0.18. In particular the marginal effect of consumption volatility is economically meaningful.
13We are grateful to Carmen Reinhart for making the crisis indicator publicly available on her website.
14Note that the crisis tally indicator is a yearly count variable. Thus we assume that if a country is in a
crisis in a given year, it is in crisis during all the months of that same year.
10
C Macroeconomic vs. Financial Risk
The previous empirical findings highlight the possibility that expected growth and macroeco-
nomic uncertainty may explain the co-movement of sovereign CDS spreads. Previous papers
have identified a link between sovereign risk premia and financial market variables such as the
variance risk premium (Wang, Zhou, and Zhou (2010)) or the CBOE S&P500 volatility index
(Pan and Singleton (2008), Longstaff, Pan, Pedersen, and Singleton (2010)). An alternative
explanation to our story would be that, as financial volatility increases, risk-averse investors
adjust their consumption patterns to account for future macroeconomic uncertainty. We
explore this hypothesis in three ways.
First, we rerun the factor regressions of equation (2) by projecting the first two factors
from the PCA on several global financial market variables such as the variance risk premium
(VRP), the CBOE S&P500 volatility index (VIX), the monthly excess value-weighted return
on all NYSE, AMEX, and NASDAQ stocks from CRSP (USret), the US price-earnings ratio
(PE), as well the difference between the Bank of America/Merrill Lynch BBB and AAA
effective yield (IG) and the difference between the BB and BBB effective yield (HY).15 We
then run a horse race with the global macroeconomic risk factors and test the robustness of
the probit specification in equation (3) against the inclusion of financial market variables.
The univariate regressions are reported in columns (3) to (14) of Table 5. Only the
coefficients for VIX, PE, IG and HY are statistically significant on their own for the first
common factor and, assuming it reflects a level factor, have economically plausible signs.
None of the coefficients are statistically significant for the second principal component.16
Moreover, for the first factor regression, the adjusted R2 of the VIX is with 55% significantly
smaller than the 75% obtained with the macroeconomic risk factors and the PE, IG and HY
have adjusted R2s in the same ballpark region. Most importantly though, for the second
15The data for the VRP is taken from Hao Zhou’s webpage, the USret from Kenneth French’s website,
the PE from Robert Shiller’s website, and the VIX, AAA BBB and BBB BB from the FRED H15 report
16Only PE is borderline significant at the 5% level.
11
factor regression, the maximum adjusted R2 of 9% is obtained for PE (4% for IG and 2% for
HY), compared to 74% for the conditional mean and volatility of US consumption growth.
A horse race between macroeconomic and financial risk confirms the former’s importance
for the common factors in the term structure of sovereign CDS spreads. Columns (1) to
(12) of Table 6 show that none of the financial market variables drives out the statistical
significance of expected growth and macroeconomic uncertainty. Again, only the PE, IG
and HY remain statistically significant for the first factor and none of them has statistical
significance for the second factor. Importantly, the sign of consumption volatility is always
preserved and changes little in magnitude. Comparing all variables in columns (13) and (14),
only the high-yield spread keeps its statistical significance at the 5% level for the level factor,
while the consumption risk factors remain highly statistically significant for both the level
and slope factors.17 Thus, our results suggest that only the high-yield bond spread (HY)
contains additional information besides the macroeconomic risk factors for the level factor
in the term structure of sovereign CDS spreads, while global financial risk has no additional
explanatory power beyond the conditional mean and volatility of consumption growth for the
slope factor. Finally, we include in column (16) of Table 5 the USret and PE into the probit
regressions as they are available since 1959. The results are qualitatively and quantitatively
maintained. In column (17), we use all financial market variables over a shorter time period
starting in 1996 and we include all countries from the Reinhart & Rogoff database. Over
the shorter time period, the coefficient of consumption volatility of 0.99 corresponds to a
marginal effect of 0.38, implying that a one percentage point rise in consumption volatility
increases the probability of a crisis by 0.38. Overall, these results corroborate our hypothesis
of U.S. macroeconomic fundamentals being a source of common sovereign credit risk.
17The coefficient on expected growth flips sign, which is due to multicollinearity issues with the PE ratio
and the investment-grade and high-yield bond spreads. We have tested that the part of expected consumption
growth orthogonal to the other variables remains statistically significant in the horse race regressions with
a negative sign on the level factor. The power and economic sign for the level factor of this orthogonalized
component is on its own interesting, but is not central to our message.
12
To close the loop of our analysis, we regress in Table 7, for each country, empirical
measures of the level and slope of the CDS term structure on the same macroeconomic
and financial risk factors.18 The level of the term structure at any day t is defined as the
average spread over all maturities, the slope is the difference between the 10-year and the
1-year spread.19 Our previous conclusions hardly change. The power of the VRP fades out
once we include the consumption risk factors. At least 58% of the coefficients on xt|t and
σt are statistically significant, while this is the case for maximally 21% of the coefficients
on the VRP. Moreover, the median adjusted R2s are high, ranging from 54% for the slope
regressions up to 70% for the level regressions. The signs of the coefficients are consistent with
the evidence of countercyclical spreads and default probabilities. A rise in expected growth
decreases the level of spreads for 68% of all countries, whereas a rise in macroeconomic
uncertainty raises the level of spreads for 97% of all countries. On the other hand, the
term structure loads positively on expected growth for more than half of all countries, and
positively on macroeconomic uncertainty for 74% of all countries in the univariate regressions.
A concern is that the level and slope measures are highly correlated. To the contrary, the
correlation between the level and slope is a weak 12%.
To summarize, our results show that global macroeconomic risk contains information un-
accounted for by financial market variables for the first two common factors in the sovereign
CDS term structure. These new findings motivate us to develop a parsimonious preference-
based model for sovereign CDSs using only two state variables, time-varying expected growth
and consumption volatility.
18We report and discuss only results for the VRP.
19Alternatively, we tried the 5-year CDS spread as a measure of the level. Results hardly change.
13
D Robustness of the Empirical Results
In order to strengthen the empirical evidence of a relationship between global macroeco-
nomic and sovereign credit risk, we conduct several robustness tests.20 First, we run the
regressions in first differences rather than in levels. Second, we discuss the use of industrial
production growth data as an alternative to consumption growth. Third, we check alterna-
tive specifications of the endowment dynamics. Fourth, we check whether our results are
robust to the inclusion of realized consumption growth. Finally, we test the relationship
between macroeconomic and financial risk in a VAR specification.
We emphasize that our analysis focuses on spread levels rather than differences, which is
consistent with Doshi, Ericsson, Jacobs, and Turnbull (2013) and references therein.21 As the
authors point out, there is no consensus in the literature, but economic intuition suggests that
spreads are mean-reverting and stationary, as opposed to trending stock prices. In addition,
first differencing comes at the cost of less efficient statistical estimates, and measurement
errors may reduce the signal-to-noise ratio more for difference regressions. We therefore
advocate the use of levels. Irrespectively of our motivation though, a concern may be that the
results are spurious because of high persistence in spreads, expected growth and consumption
volatility. Therefore, we have reported results using difference regressions in section A-V of
the internet appendix. Consistent with the literature, R2 statistics are significantly smaller
for the difference regressions, ranging around 9%. But we maintain statistical significance,
in particular for the regressions with the slope factor. Importantly, we run a Dickey-Fuller
test on the residuals of the regressions in levels and we strictly reject the presence of a unit
root (see last row in Tables 5 and 6). Regressions are thus not spurious. Even if the factors
and consumption variables were integrated of order one, these results would suggest that
there exists a cointegrating relationship. We don’t pursue such tests as they are not the
20We thank an anonymous referee for suggesting several of these robustness checks.
21Campbell and Taksler (2003) use spread levels, and Benzoni, Collin-Dufresne, Goldstein, and Helwege
(2012) advocate regressions in levels.
14
focus of our study. However, if we were to find evidence in favor of a long-run equilibrium
relationship, this would still not undo our message that there exists a strong relationship
between the common information in the term structure of spreads and macroeconomic risk.
Next, we verify our results using monthly industrial production growth data instead of
monthly consumption growth, similar to Bansal, Kiku, Shaliastovich, and Yaron (2013),
Joslin, Le, and Singleton (2013) and Lustig, Roussanov, and Verdelhan (2013). Industrial
production growth data has the advantage of being available since 1919 and correlates sub-
stantially with consumption growth at lower frequencies. Unreported results suggest that
we get statistically significant estimates in the regression analysis with heteroscedasticity-
robust, but not with block-bootstrapped standard errors. We believe that this may be due
to the fact that industrial production, which accounts for inputs and outputs of both non-
durables and durables, is an imperfect proxy for the non-durable component of consumption
growth. At the monthly horizon from 1959 to 2010, the correlation coefficient between the
two time series is only 19.88%.
We further test whether our base regression results are affected by the functional form
specification of consumption volatility. We adopt other GARCH models commonly adopted
in the financial literature: the standard GARCH(1,1) model (Bollerslev (1986)), the EGARCH
model (Nelson (1991)) and the GJR GARCH model (Glosten, Jagannathan, and Runkle
(1993)). The dynamics and parameter estimates of the different GARCH specifications are
reported in Panel B of Table 4. These alternative specifications are comparatively more
persistent, with estimates for φσ ranging from 0.9779 for the GARCH(1,1,) specification to
0.9907 for the EGARCH specification. Replications of the benchmark regression results,
using these different volatility specifications, are reported in section A-VI of the external
internet appendix because of space restrictions. Both the univariate and the multivariate
tests illustrate that a different functional form of the volatility process does not alter our
conclusion of a strong relationship between the first two principal components and expected
consumption growth and volatility.
15
GARCH models inherit contemporaneous correlation between realized consumption growth
and consumption volatility. Hence there is a possibility that large moves in realized con-
sumption growth mechanically lead to large increases in volatility. Therefore we test whether
the significance of the macroeconomic risk factors survives once we control for variation in
realized consumption growth. Unreported results suggest that the statistical relationship
between consumption volatility and the common information in the term structure of CDS
spreads is not merely due to jumps in realized consumption growth. The regression coeffi-
cient on realized consumption growth is insignificant and neither the statistical significance
nor the magnitude of the coefficient on expected growth and consumption volatility changes.
Finally, we investigate the relationship between the macroeconomic risk factors and the
VIX index in a VAR specification. Unreported results show no evidence that financial market
volatility Granger causes expected growth. Moreover, results between consumption volatility
and the VIX are inconclusive and point to mere correlation.
III A Macroeconomic Model for Sovereign CDS
Our empirical exercise highlights that expected U.S. consumption growth and macroeco-
nomic uncertainty accounts for a large fraction of the co-movement in sovereign CDS spreads.
We show that an equilibrium model for CDS spreads with only two state variables can ratio-
nalize these facts. Our ingredients are an endowment economy with time-varying expected
growth and consumption volatility, recursive Epstein and Zin (1989) preferences, and an
embedded reduced-form default process driven by the two state variables of the economy.
A Credit Default Swaps
To derive the valuation of CDS spreads in closed-form, we discretize the continuous frame-
work in Duffie (1999), albeit adapting the explicit modeling of the hazard and recovery rate.
We write the model at a daily frequency in order to agree with daily quotations in the CDS
16
market. We assume that each coupon period n contains J trading days.22 A K-period CDS
thus has KJ trading days. CDSs are priced similar to interest rate swaps, that is expected
net present values of cash flows for both legs (protection buyer and protection seller) must
equalize at inception. For a K-period CDS, the expected net present value of the protection
leg, πpbt , to be paid by the protection buyer is equal to
(4)
πpbt = CDSt (K)
(K∑k=1
Et [Mt,t+kJI (τ > t+ kJ)] + Et
[Mt,τ
(τ − tJ−⌊τ − tJ
⌋)I (τ ≤ t+KJ)
]),
where CDSt (K) is the constant premium agreed at day t and to be paid until the earlier
of maturity (day t + KJ), or a credit event occurring at a random day τ . For t′ > t, Mt,t′
denotes the stochastic discount factor valuing any financial payoff to be claimed at a future
day t′. The floor function b·c rounds a real number to the largest previous integer, and I (·)
is an indicator function taking the value 1 if the condition is met and 0 otherwise. The first
part in equation (4) defines the net present value of payments made by the protection buyer
conditional on survival. The second part defines the accrual payments if the reference entity
defaults between two payment dates.
The protection seller must cover any losses incurred by the protection buyer in case of a
credit event. The expected net present value of the protection seller’s leg, πpst , is equal to
(5) πpst = Et [Mt,τ (1−R) I (τ ≤ t+KJ)] ,
where R represents the constant post-default recovery rate as a fraction of face value, and we
define the constant Loss Given Default L = (1−R).23 We fix the dynamics of the recovery
rate process at a constant level, consistent with industry standards for CDS pricing. This
22Note that the period n contains the trading days (n− 1) J+ j, j = 1, . . . , J . In the calibration exercise,
we assume without loss of generality that swap premia are paid on a yearly basis. The assumption of yearly
payments assures that the model results can directly be translated into annualized spreads. However, the
model can easily accommodate bi-annual and quarterly payment frequencies.
23In what follows, we will interchange freely between the notions of Loss Given Default and Loss Rate.
17
is also in line with standard assumptions in the CDS pricing literature such as Pan and
Singleton (2008) or Longstaff, Pan, Pedersen, and Singleton (2010). We leave the analysis
of time-varying recovery rates for further research.24
Equating the two legs, as the expected net present values are zero at inception, and
applying the Law of Iterated Expectations, we can write the CDS spread as
(6) CDSt (K) =
KJ∑j=1
Et [Mt,t+j (1−R) (St+j−1 − St+j)]
K∑k=1
Et [Mt,t+kJSt+kJ ] +KJ∑j=1
(jJ − b
jJ c)Et [Mt,t+j (St+j−1 − St+j)]
,
where the survival probability St ≡ Prob (τ > t | It) denotes the conditional probability that
the credit event did not occur at day t, and where It denotes the information set up to and
including day t.25 The conditional survival probability St is defined as
St = S0
t∏j=1
(1− hj) , t ≥ 1,(7)
where the hazard rate process ht ≡ Prob (τ = t | τ ≥ t; It) denotes the conditional instanta-
neous default probability of a given reference entity at day t.
To derive analytic solutions to the CDS, we need to specify dynamics for the stochastic
discount factor Mt,t+1 and the default intensity ht+1. These processes are determined by the
two state variables of the economy, which are described in the following section.
B A Markov-switching Model for Consumption Growth
In section II, we estimate the dynamics of consumption growth using a continuous-state
space model because a discrete approximation with regimes does not provide a sufficiently
good approximation in small samples for highly persistent processes such as expected con-
24We did explore the possibility of deterministic procyclical and state-dependent recovery rates, while
keeping the unconditional recovery rate fixed. Our results are not very sensitive to such a variation.
25Note that we assume Prob (τ = t | It′) = Prob(τ = t | Imin(t,t′)
)for all integers t and t′.
18
sumption growth and macroeconomic uncertainty. For the purpose of our model, we do
however approximate the continuous-state dynamics with a discrete regime-switching frame-
work as it is well known that Markov switching models provide excellent approximations of
continuous processes in population (Timmermann (2000), Bonomo, Garcia, Meddahi, and
Tedongap (2011)). More importantly, without the discrete-state approximation, we are not
able to obtain analytical formulas for spread moments in our preference framework, which
is particularly useful for computational efficiency when we want to estimate the preference
parameters. In addition, if we limit us to two regimes for each state variable, the closed-form
formulas enhance the understanding and interpretation of the mechanisms underlying the
empirical results.
Following Bonomo, Garcia, Meddahi, and Tedongap (2011), we assume that both the
mean and variance of consumption growth gt+1 fluctuate according to a Markov variable st,
which can take a different value in each of the N states of the economy. The stochastic
sequence st evolves according to a transition probability matrix P
(8) P> = [pij ]1≤i,j≤N , pij = Prob (st+1 = j | st = i) .
As in Hamilton (1994), let ζt = est , where ej is the N × 1 vector with all components equal
to zero but the jth component equals one. Formally, consumption growth can be written as
(9) gt+1 = xt + σtεg,t+1,
where xt = µ>g ζt and σt =√ω>g ζt are the expected mean and the volatility of consumption
growth respectively. The vectors µg and ωg contain the values of expected consumption
growth and consumption volatility respectively in each state of the economy, and the com-
ponent j refers to the value in state st = j. For simplicity, we limit the number of states to
two for each risk factor, indexed by the letters L for the low state and H for the high states.
This amounts to a total of four states (LL, LH, HL and HH) if the factors are linearly
independent.
19
C Preferences and Stochastic Discount Factor
We study the valuation of CDSs in the context of a representative agent general equilibrium
model. We assume that the representative investor has Epstein and Zin (1989) preferences.
Such an investor derives lifetime utility Vt recursively from the combination of the current
level of consumption Ct and Rt (Vt+1), a certainty equivalent of next period lifetime utility
Vt =
{(1− δ)C
1− 1ψ
t + δ [Rt (Vt+1)]1− 1
ψ
} 1
1− 1ψ if ψ 6= 1(10)
= C1−δt [Rt (Vt+1)]δ if ψ = 1,
where Rt (Vt+1) =(Et[V 1−γt+1
]) 11−γ and where δ represents the subjective discount factor.
The parameter ψ defines the elasticity of intertemporal substitution (EIS), which can be
disentangled from the coefficient of relative risk aversion γ through this form of utility.
Hansen, Heaton, and Li (2008) derive the stochastic discount factor Mt,t+1 in terms of
the continuation value of utility of consumption as
(11) Mt,t+1 = δ
(Ct+1
Ct
)− 1ψ(
Vt+1
Rt (Vt+1)
) 1ψ−γ.
If γ = 1/ψ, equation (11) corresponds to the stochastic discount factor of an investor with
time-separable utility and constant relative risk aversion. Alternatively, if γ > 1/ψ, Bansal
and Yaron (2004), for instance, show that a premium for long-run consumption risk is added
by the ratio of future utility Vt+1 to its certainty equivalent Rt (Vt+1).
Based on the dynamics (9) and the recursion (10), we show in the internet appendix
section (A-II.A) that the stochastic discount factor (11) may be expressed as
Mt,t+1 = exp(ζ>t Aζt+1 − γgt+1
),(12)
where the components of the N×N matrix A are described in equation (A-9) of the appendix.
We assume that the representative agent is based in the U.S. and that he values his
international investment opportunity set using the domestic pricing kernel. This assumption
20
is merely for simplicity and is in no way restrictive. Borri and Verdelhan (2009) theoretically
prove that as long as markets are complete and assets are USD denominated, all results carry
through for non-U.S. investors who use their foreign currency denominated discount factor.
The same results apply to our set up as we are pricing only USD denominated CDS.
D Hazard Rate
Given our intention to link the default process to global macroeconomic risk, we assume that
the hazard rate hit for each country i follows a logistic distribution26
(13) hit =λit
1 + λitwhere λit = exp
(βiλ0 + βiλxxt + βiλσσt
),
where the default intensity λit is impacted by expected growth and consumption volatility.
Heterogeneous exposures to the macroeconomic risk factors are determined through the pa-
rameters βiλx and βiλσ, and βiλ0 is a country-specific constant. We will drop the i superscript
for ease of readability.27 This set up guarantees that the hazard rate is well defined and
belongs to the interval (0, 1). In addition, it ensures that the marginal propensity to de-
fault is persistent, given that expected consumption growth and volatility are themselves
persistent processes. While the model-specific investor preferences are needed to generate
sufficiently large countercyclical risk aversion necessary to match the levels of CDS spreads,
a persistent and time-varying default intensity, coupled with a preference for early resolution
of uncertainty, is essential to generate a term risk premium.
Sovereign default risk is modeled as a reduced-form process, where the latent default
intensity depends on the global macroeconomic risk factors exogenously specified in the
26This functional form of the hazard rate is not essential to derive closed-form expression for CDS spreads,
nor for our quantitative results. Our results are robust to a Weibull-type hazard rate, specified as ht =
1− exp (−λt) ,where λt = exp (βλ0 + βλxxt + βλσσt).
27An alternative to this approach is to define a representative country and model its rating transition
probability matrix. We pursue this methodology in parallel work, but it is not the focus of our analysis here.
21
endowment economy. This is similar in spirit to models that link the default intensity to
observable covariates, such as in Duffie, Saita, and Wang (2007) for default prediction and
in Doshi, Ericsson, Jacobs, and Turnbull (2013) to model the dynamics of CDS spreads. We
choose to embed a reduced-form credit risk model into an equilibrium set up for two reasons.
First, a CDS payout is triggered through a credit event. Longstaff, Pan, Pedersen, and
Singleton (2010) point out that Bankruptcy does not figure among the spectrum of credit
events for sovereign contracts. This is due to the nonexistence of a formal international
bankruptcy court for sovereign borrowers. As we focus on the pricing of CDS spreads prior
to default, we are less interested in the default decision as such, but rather in sovereign
distress risk (abstracting from the ”jump-at-default” risk in the literature).28 Our set up
thus differs from the endogenous default literature starting with Eaton and Gersovitz (1981),
which typically imposes exogenous default costs, which are also necessary to justify optimal
default decisions in bad states (Arellano (2008)). Recent work by Mendoza and Yue (2012)
provides a foundation to justify these exogenous default costs and highlights the stylized
fact of countercyclicality in credit spreads (and thus default probabilities) in a business cycle
model with endogenous default.
Second, a government’s default decision is mostly strategic. Modeling an explicit default
threshold in a structural default risk model for a sovereign requires therefore additional
assumptions (such as ad-hoc exogenous default costs, for instance). Structural default models
for corporate default have successfully been integrated into equilibrium models by Chen,
Collin-Dufresne, and Goldstein (2009), Chen (2010) and Bhamra, Kuehn, and Strebulaev
(2010). The success of these models relies crucially on the ability to generate countercyclical
default probabilities. This property is obtained in our reduced-form framework if the default
intensity increases when expected growth is low and/or macroeconomic uncertainty is high.
This is guaranteed as long as βλx and βλσ are non-positive and non-negative respectively.
We emphasize that our goal is to show how a simple equilibrium model with only two
28For the rest of the paper, we will refer more generally to the term default risk.
22
macroeconomic state variables can rationalize several stylized facts of the sovereign CDS mar-
ket. We are aware that our specification implies that heterogeneity in the level of spreads
arises only because of differential sensitivity to aggregate risk. While idiosyncratic shocks
may play no role in CDS returns, they may be important for the variation in spread levels.29
We therefore derive solutions where the hazard rate contains additional country-specific
shocks. The results improve only marginally, but the number of states increases to eight,
even if we allow only for two states of the idiosyncratic Markov chain. This renders the inter-
pretation of the state-dependent results cumbersome. We therefore focus on the aggregate
risk factors only and provide the additional formulas and results in the internet appendix.
Using our pricing framework and specification of the default process, we derive in internet
appendix section (A-II.B) the conditional and unconditional cumulative default probabilities
over a (T − t)-year horizon
Probt (t < τ ≤ T | τ > t) = 1−(
Ψ>T−tζt
)(14)
Prob (t < τ ≤ T | τ > t) = 1−(
Ψ>T−tΠ),
where the maturity-dependent vector sequence{
Ψj
}satisfies the internet appendix recursion
(A-12) with initial condition Ψ0 = e, where e is the vector with all components equal to
one, and where Π denotes the vector of unconditional state probabilities. We complement
the historical cumulative default probabilities with closed-form solutions of the cumulative
default rate under the risk-neutral (Q) measure in internet appendix section (A-II.C).
E Credit Default Swap Spread
Markov chains are crucial for deriving analytical solutions for CDS spreads and their mo-
ments. We develop equation (6) in appendix (A-II.D) to characterize a K- period CDS
(15) CDSt (K) = ζ>t λs (K) ,
29We thank an anonymous referee for pointing this out.
23
where the components of the vectors λs (K) are non-linear functions of the consumption
dynamics, the default process, the recovery rate and of the recursive utility function. Its
components are given by
(16) λi,s (K) =
KJ∑j=1
L[Ψ∗i,j −Ψi,j
]K∑k=1
Ψi,kJ +KJ∑j=1
(jJ −
⌊jJ
⌋) [Ψ∗i,j −Ψi,j
] ,
where L is the vector of loss rates, and where the sequences{
Ψ∗j}
and {Ψj} are given by
the internet appendix recursion (A-30), with initial conditions (A-29). All moments of CDS
spreads exist in closed form, which is particularly useful for the estimation of the default
and preference parameters, given the highly non-linear pay-off function of CDS spreads.
IV Model Estimation
We first describe the calibration for the dynamics of aggregate consumption growth, which
is the only exogenous process in the model. We then explain how we estimate the structural
preference parameters and the cross-sectional sensitivities of the default process to aggregate
risk factors by exploiting the high-frequency information in daily CDS spreads.
A Consumption Growth Dynamics
We calibrate the consumption growth dynamics at a monthly frequency as in Bansal, Kiku,
and Yaron (2012) to match observed annual growth rates from 1930 to 2008, because we
target a longer sample of consumption data than the monthly post-war data available only
since 1959.30
30The calibrated parameters differ slightly from the Kalman Filter estimates obtained in the empirical
section. Nevertheless, all calibrated parameters for expected growth remain within the tight bound of one
standard deviation away from the estimates. Moreover, the persistence of consumption volatility is within
two standard deviations of the GARCH, EGARCH and GJR-GARCH volatility estimates.
24
The mean expected consumption growth µx is 0.0015 with a sensitivity to long-run risk
shocks νx set to 0.038. These shocks are persistent with a value φx equal to 0.975. Con-
sumption volatility has a mean√µσ of 0.0072, a persistence φσ of 0.999 and a sensitivity to
shocks νσ of 2.80×10−6. As we want to exploit the rich information in the daily dynamics of
CDS spreads to estimate the structural default and preference parameters, we choose to map
the monthly calibrated values into a daily frequency, assuming twenty-two trading days in a
month. More specifically, we translate the monthly dynamics of consumption growth into a
daily system such that the time-averaged first and second moments of annual consumption
growth are preserved in population. For the purpose of illustration, a value of µx = 0.0015
at a monthly decision interval for the mean consumption growth corresponds to a value at
a daily decision interval equal to µdailyx = ∆µx, where ∆ = 1/22. Similarly, a value of φx =
0.975 for the persistence of the predictable component of consumption growth is translated
into a daily value equal to φdailyx = φ∆x (see Table 8 for further details).
The mapping from the continuous daily endowment process into a discrete Markov-
switching model follows the procedure described in Bonomo, Garcia, Meddahi, and Tedongap
(2011). Calibration results for the consumption growth dynamics are reported in Panel A
of Table 8, where we obtain values for all four states of nature defined by the combinations
of low (indexed by the letter L) and high (indexed by the letter H) conditional means and
variances of consumption growth. Panel B reports annualized (time-averaged) descriptive
statistics for aggregate consumption growth, which are compared against the observed values.
The mean, volatility and persistence for consumption growth of 1.8%, 2.53% and 0.46 are
consistent with the estimates of 1.92%, 2.12% and 0.46 found in the data.
B Estimation of Preference and Default Parameters
We exploit the high-frequency information in daily sovereign CDS spreads to estimate the
preference parameters as well as the cross-sectional sensitivities of the default process to the
two systematic risk factors via the Generalized Method of Moments (GMM). The weighting
25
matrix is the inverse of the diagonal of the spectral density matrix, and the moments are the
expectations of CDS spreads and their squares. We thus have 72 moments for 36 spread series
(6 maturities for each rating category) to estimate in total 21 parameters, i.e. 3 preference
parameters and 3 default parameters for each of the 6 rating groups.31 The recovery rate
is fixed at 25%, which is the standard recovery rate for sovereigns and also consistent with
Pan and Singleton (2008) and Longstaff, Pan, Pedersen, and Singleton (2010).
Estimation results with Newey-West standard errors are reported in Table 9. We fix
the subjective discount factor δ at a monthly frequency at 0.9989 and estimate the other
preference parameters. All coefficients turn out to be statistically significant at the 1%
level. Using spread levels (changes) for the estimation, we obtain a coefficient of relative risk
aversion γ of 8.2692 (7.1713). This is significantly less than the value of 20.90 estimated in
Bansal and Shaliastovich (2013), who estimate a value of 1.81 for the EIS ψ at a quarterly
model frequency. Part of the discrepancy could be due to a different decision interval, as
time-aggregation can lead to higher estimates of risk aversion. We estimate a value of the
EIS equal to 1.5774 (1.5953). The daily information in CDS spreads provides additional
support for a value of the EIS above 1 and preference for early resolution of uncertainty.
Regarding the default parameters, all signs of the coefficients are consistent with economic
intuition and imply countercyclical default probabilities.32 This feature is essential for asset
pricing implications as emphasized in Bhamra, Kuehn, and Strebulaev (2010), among others,
and helps us generate countercyclical credit spreads. Negative values for βλx suggest that a
rise in expected consumption growth lowers the marginal propensity to default. Moreover,
positive values for βλσ indicate that in times of high macroeconomic uncertainty, sovereign
debt becomes riskier and the likelihood of default increases. Thus, asset markets dislike
macroeconomic uncertainty (Bansal, Khatchatrian, and Yaron (2005)). Also, the average
default intensity is inversely related to credit-worthiness, that is the constant coefficient βλ0
31A detailed description of the estimation steps is provided in internet appendix section (A-II.E).
32Default parameter estimates using spread changes are available upon request.
26
increases from negative 15.37 to negative 9.15 for the AAA and B rating category respectively.
Furthermore, highly rated countries are more sensitive to macroeconomic uncertainty and
the pattern is monotonically decreasing, while there is no clear pattern for the sensitivity
to expected consumption growth. The last panel reports the J-statistic for the test of
overidentifying restrictions and the corresponding p-value. The model is not rejected at the
1% significance level.
V Asset Pricing Implications and Discussion
We first study the model implications for unconditional CDS moments and default proba-
bilities. Then, we investigate asset pricing results for the conditional CDS term structure.
A more detailed analysis of the hazard rate is followed by an extension of the model to
preferences that incorporate generalized disappointment aversion.
A Unconditional CDS Moments
All results for the unconditional moments of the CDS term structure are summarized in
Table 10 in the left panel. We evaluate the outcome against empirical moments in the data
reported in the right panel based on the root mean squared error (RMSE)
(17) RMSE =
√√√√ 1
K
K∑j=1
(xj − xj)2,
where K represents the CDS contract maturity, xj is the unconditional model-implied mo-
ment and xj refers to the empirical counterpart.
The model does a particularly good job in reproducing the unconditional mean, standard
deviation, skewness and first-order autocorrelation coefficients of the CDS term structure.
We generate consistently a mean upward sloping term structure for all rating categories,
in line with the data. This is consistent with the finding of persistent upward sloping term
27
structures for Mexico, Turkey and Korea reported by Pan and Singleton (2008).33 RMSEs for
the term structure of spreads are approximately 1 basis point for investment grade reference
entities, and range from 4 to 14 basis points for high-yield categories. To give an example, for
single A rated countries, the one-year (ten-year) spread is 37 (73) basis points against 35 (71)
in the data. We also fit the upward sloping term structure of volatility for groups AAA to A,
and the downward sloping volatility pattern of groups BBB to B. RMSEs range from 1 to 2
basis points for investment-grade countries, and are less than 12 basis points for high-yield
countries. Only for the B group, the RMSE is 27 basis points, implying an average error
of roughly 10% for the 5-year B volatility. In addition, the model is highly satisfactory in
reproducing the right-skewed distribution of CDS spreads and the first-order autocorrelation
coefficients of the observed spreads, which are highly persistent. While the model generates
excessive leptokurtic distributions at short maturity contracts, it reproduces the pattern
of decreasing fat tails with asset horizon and converges to the observed results at longer
maturities. We also note that the average level of spreads is higher for less-creditworthy
countries, which is consistent with the monotonic pattern in the coefficient estimates of βλ0.
In our model, time-varying global macroeconomic risk feeds into the default process.
Negative shocks to expected consumption growth and macroeconomic uncertainty are per-
sistent and create uncertainty about future default rates, which is priced. In addition to a
level risk premium, preference for early resolution of uncertainty helps to generate a term
premium, which rises with the asset horizon. In sum, with only two macroeconomic state
variables, we are able to match closely the term structure of spreads, volatility, skewness and
kurtosis, as well as the persistence of CDS spreads across 6 broad rating categories.
Before analyzing the conditional asset pricing implications and default probabilities, we
highlight that, given the calibrated endowment dynamics and estimated preference param-
eters based on sovereign CDS spreads, we remain consistent with the stock market. The
33Every country in our sample has an upward sloping term structure on average. Because of space
limitations, we have only reported summary statistics for the aggregated series in Table 2.
28
model generates a sizable equity premium of 5.52%, equity volatility of 17.48%, a risk-free
rate of 1.01% with a volatility of 0.80% (see Table 8). Thus, we provide a joint framework to
price stocks and CDS. A strong overlap in the stochastic discount factor for pricing both the
U.S. equity market and the global sovereign CDS market suggests that both are integrated
and reflects previous evidence of information flow between the two asset markets.34
B Default Probabilities
Model-results for cumulative default probabilities under the physical and risk-neutral mea-
sure, as well as their ratios, are reported in Table 11. We benchmark the model outcomes
against the historical sovereign foreign-currency cumulative average default rates reported
by Standard&Poor’s over the time period 1975 to 2009.35 Inspection of these numbers war-
rants several explanations at the outset of our discussion. Both physical and risk-neutral
default probabilities are unobserved and a proper comparison is thus close to impossible. In
particular, no country rated A or higher has defaulted within the last 40 years.36 Although
arguably very small, the default probability of a AAA rated country is unlikely zero. This
is particularly true for the CDS market, where a technical default could trigger a payout.
Therefore, we use cumulative historical default probabilities from the cash market as a first
best benchmark for comparison, but we remain critical about their use.
Given the low RMSEs, the model-implied default probabilities for CDS under the phys-
ical measure are close to their observed counterparts from the cash market. The lowest and
highest RMSEs are 0.97% and 11.11% for AAA and B rated countries respectively. Cumula-
tive default probabilities line up cross-sectionally, with the 5-year default probability ranging
from 0.79% to 22.60% for the AAA and B rated entities. Moreover, cumulative default prob-
34See Acharya and Johnson (2007) among others.
35The results benchmarked against the cumulative default rates reported by Moody’s are very similar.
36While no A rated country had an outright default, countries may be downgraded prior to default. We
study rating migrations in parallel work.
29
abilities rise with the asset horizon, which is consistent with increasing cumulative default
probabilities over time. Results are slightly better at longer than at shorter horizons.
The ratios of risk-neutral to physical default probabilities are monotonically increasing
with maturity, reflecting the term premium required for selling CDS insurance. Similar
for the physical default probabilities, the risk-neutral values are cross-sectionally increasing
for less credit-worthy countries. All ratios of risk-neutral to physical probabilities range
between 1.09 and 3.98, while the average is 2.15. These values are consistent with Berndt,
Jarrow, and Kang (2007), who find strongly time-varying ratios of implied risk neutral
default probabilities to Moody’s KMV Expected Default Frequencies between 1 and 3 for
short horizons, but going as high up as 10 in 2002. For corporate bonds, Driessen (2005)
and Huang and Huang (2012) who find average ratios of, respectively, 1.89 and 1.11 to 1.75.
C Conditional CDS Moments
The regime-switching set up of the model allows for a better understanding of the relationship
between macroeconomic risk factors and asset prices. In Figure 3, we report state-dependent
spreads for the four states of nature determined by expected growth and volatility of con-
sumption, as well as the unconditional moments.
We first note that, ceteris paribus, an increase in expected consumption growth shifts the
entire term structure downwards. These shifts are visible, for instance, when moving from
the solid to the dashed line, and from the dotted to the dash-dotted line. In contrast, a rise in
consumption volatility has the opposite effect on the level of spreads. Higher macroeconomic
uncertainty raises the level of spreads. This model-implied result is consistent with the
empirical regression results. More specifically, the projection of the first principal component,
extracted from the term structure of spreads across all countries, on the two macroeconomic
risk factors yields, respectively, a negative and positive loading on expected growth and
consumption volatility. In addition, we remind that the average country loadings on the first
principal component are maturity-invariant and uniformly positive across reference entities.
30
Thus if we believe that the first principal component is a level factor in the term structure
of spreads, then a negative coefficient a1 on expected growth in equation 2 implies that the
level of sovereign CDS spreads is lower in states of high conditional expected consumption
growth. Moreover, a positive coefficient on a2 implies that higher volatility of aggregate
consumption growth increases sovereign spreads. These results are economically intuitive
and consistent with a countercyclical level of spreads. We add that this relationship between
the level of spreads and the two macroeconomic risk factors is also obtained in the country
regressions reported in Table 7, where we use the actual level of spreads as the regressand.
Replicating these regressions inside the model, we find that the model-implied results in
population predict a negative and positive coefficient, respectively, on expected growth and
consumption volatility for the level of spreads.37 The theoretical R2 of 97% of that regression
compares favorably to the empirical value of 75%.
While global macroeconomic risk affects the level of spreads, this effect is asymmetric
across maturities and thereby affects the slope of spreads. Irrespectively of whether consump-
tion volatility is high or low, a positive growth outlook steepens the CDS term structure.
This is because a positive contemporaneous shock to expected consumption growth will de-
crease default probabilities, given the negative estimates of βλx. The improved likelihood
of default is particularly pronounced at shorter horizons, since at longer horizons, there is
greater uncertainty and a higher probability of negative shocks to expected growth (and
therefore higher default probabilities). Given preference for early resolution of uncertainty,
this commands a term premium, which increases the slope of the term structure. This model-
implied feature is also reflected in the positive slope coefficient a1 from the regression of the
second principal component on expected consumption growth reported in regression (2) of
Table 5, if we are to interpret the second principal component as a slope factor. It is likewise
37Observe that in the model, the Variance Risk Premium is perfectly collinear with expected consumption
growth and volatility. Comparing the model predictions for the slope coefficients and R2 to the empirical
observations when all factors are included in the specification is thus not possible.
31
consistent with the country regressions, where more than half of the regression coefficients
for the slope are positive. Columns (10) to (12) of Table 7 show that, in the model, expected
growth positively predicts the slope in population with a R2 of 75%, which squares well with
a value of 74% and a positive regression coefficient in the second factor regression.
The relationship between consumption volatility and the slope depends on the perception
of future economic conditions. If expected consumption growth is low, higher macroeco-
nomic uncertainty will decrease the slope of the term structure. However, conditional on
high expected consumption growth, the term structure steepens following a positive shock to
macroeconomic uncertainty. The positively estimated hazard rate coefficients βλσ indicate
that a rise in uncertainty increases default probabilities and therefore raises spreads. In times
of high expected growth, if the agent expects more volatility in the future, with preference
for early resolution of uncertainty, he will command a term premium because of the aversion
towards uncertainty shocks. This increases the slope. However, if expected growth is low,
a positive volatility shock pushes the marginal investor into the worst economic state and
induces a strong deterioration of conditional default probabilities. Conditional on survival,
default probabilities are expected to be lower at longer horizons and this effect dominates
over a term risk premium. Unconditionally, we expect the positive relationship between un-
certainty and the slope of spreads to dominate as in our model the unconditional probability
of being in a state of high expected consumption growth (≈89%) is larger than the proba-
bility of being in a state of low expected growth (≈11%). Indeed, this outcome is predicted
by the model-implied replication of the country regressions in columns (10) to (12) of Table
7. This explanation also rationalizes the positive regression coefficient a2 in column (2) of
Table 5 if we believe the second principal component to be a slope factor.
Figure 3 further emphasizes that the slope of the term structure inverts in times of low
expected growth and high macroeconomic uncertainty. The economic magnitude of the
inversion ranges from 9 to 93 basis points, in absolute value, for the AAA and B rating
categories respectively. Thus, an investor becomes much more risk averse in bad economic
32
times and requires higher compensation to offer protection on sovereign default. This jacks
up the price of CDS at short horizons. The possibility of reverting back to sound economic
fundamentals introduces mean reversion and spreads converge back to lower levels at longer
horizons. But as shocks are persistent, the average level remains elevated.
For the purpose of comparison, we plot in Figure 4 the historical difference between the
10 and 1-year CDS spread of four European distressed economies during the sovereign debt
crisis in the north-east corner of the figure.38 In the year prior to default, Greece was rated
B. Within our sample, the slope of its CDS term structure inverted by a maximum of 525
basis points and the average slope across reversal dates is 195 basis points. This is about
twice as large as the inverted spread curve of 93 basis points generated by the model in the
worst state. In addition, Portugal, which was rated A in 2010, had a maximum reversal
of approximately 150 basis points and an average inverted slope of 68 points, which is also
a bit more than twice the conditionally inverted slope of 26 basis points. Moreover, the
mean (highest) slope reversal for Spain, which was downgraded to AA+ in 2010, was 29
(56) basis points, which again is twice the negative slope of 14 basis points. The slope of
Italy didn’t decrease as much during our sample period, but it continued to decrease as the
sovereign debt crisis intensified. In sum, the conditional moments of our model qualitatively
match the slope reversal of distressed sovereign borrowers in states of bad macroeconomic
fundamentals. We quantitatively underestimate the inversion by about 50%.
D An Analysis of the Hazard Rate
To provide further intuition about how the effects of the systematic risk factors affect the
term structure, we plot in Figure 5 the results of our benchmark model against different
specifications of the hazard process from equation 13. The solid bullet points are the un-
conditional moments from the data and the dashed-dotted line represents the unconditional
moments when both expected consumption growth and volatility are active. With a constant
38Graphs for other rating categories than AA, A and B look very similar and are available upon request.
33
default process (dashed line), the term structure is flat. This is not surprising, as default
intensities become deterministic, and we expect no term premium. On the other hand, a
specification with only expected consumption growth (dotted line) creates a term structure,
which is way too steep, while the opposite is true, when consumption volatility is the only
risk driver. Thus, the first and second moments of aggregate macroeconomic risk have offset-
ting effects on the term structure, and a two-factor model is necessary to reproduce stylized
facts observed in the data. This compares with Pan and Singleton (2008) and Longstaff,
Pan, Pedersen, and Singleton (2010) for example, who use a one-factor model. However, the
authors argue that a one-factor model is acceptable, but that a two-factor model may be
desirable.
E An Extension to Preferences with Downside Risk Aversion
Our model is successful in matching the unconditional moments of the term structure, but
we quantitatively underestimate the inversion of the term structure in the worst economic
state. We try to improve upon this dimension by extending our benchmark scenario with
symmetric recursive preferences and a Kreps and Porteus (1978) (KP) certainty equivalent
to that of a risk averse investor with generalized disappointment averse (GDA) preferences of
Routledge and Zin (2010).39 This framework is well suited as GDA preferences endogenously
generate higher risk aversion in the worst economic state. The extended model derivations
are relegated to the external appendix. Beside the subjective discount factor δ, the coef-
ficient of relative risk aversion γ and the elasticity of intertemporal substitution ψ, GDA
preferences are characterized through two additional preference parameters. The coefficient
of disappointment aversion, α, defines the weight attributed to disappointing outcomes. A
state-contingent outcome is only disappointing if it falls below a fraction κ of the certainty
equivalent. We first discuss the estimation results, then the asset pricing implications.
Table 9 shows that the additional degree of freedom slightly improves the fit of the
39In fact, the GDA preferences of Routledge and Zin (2010) nest those of Epstein and Zin (1989).
34
model. The model is not rejected and the J-Test statistic is a bit lower than in the KP
case with a value of 14.65. A likelihood ratio test rejects the benchmark model against
the alternative with generalized disappointment averse preferences.40 Fixing the subjective
discount factor at 0.9989, we find that all coefficients are statistically significant at the 1%
level. Using spread levels (changes), we estimate a coefficient of relative risk aversion γ of
4.9081 (5.0614) and an EIS ψ equal to 1.4874 (1.2473). Similar to the benchmark case, this
implies a value for the EIS above 1 and preference for early resolution of uncertainty. The
parameter of disappointment aversion α is equal to 0.2486 (0.4819), implying that the ratio
of investor’s marginal utility of wealth for non-disappointing to disappointing outcomes is
25% (48%).41 The parameter κ, which measures the fraction of the certainty equivalent
below which disappointment kicks in, is estimated to be 0.8470 (0.9549). Estimation of the
generalized disappointment averse preference parameters using derivative pricing information
in the fixed income market is a novel result in the literature.
Table 12 illustrates that the performance of the KP and GDA scenarios are similar for
the unconditional moments of the CDS term structure. The GDA model performs slightly
better for the mean term structure, with RMSEs ranging between 0.52 and 9.60 basis points
respectively. On the other hand, the model generates comparable results for the volatility,
skewness and persistence of CDS spreads. In addition, it introduces slightly higher kurtosis
at the short end of the curve for the asset distributions.
Figure 3 further illustrates that the downside risk aversion steepens the inversion of
the term structure in states of low expected consumption growth and high macroeconomic
uncertainty. Asymmetric preferences thus improve the fit in the worst economic state if we
compare the model-implied results against the average term structure inversion of rating-
equivalent countries in distress during the sovereign debt crisis, as we do in Figure 3 for the
40For the estimation using spread changes, we also reject the KP model in favor of GDA preferences.
41A value of α equal to 0.25 maps into a loss aversion parameter of 3, a well accepted measure in the
literature for loss aversion.
35
benchmark case. An investor who cares about downside risk jacks up the price of credit
protection much more in the worst economic state. The magnitude of the slope reversal
in the bad state is consistently about twice the value obtained with symmetric preferences.
The 10 minus 1 year slope decreases from -17 basis points for AAA countries, to a minimum
of -169 basis points for the worst rating category B. We thus find that both the KP and the
GDA economies manage to match the unconditional moments of the CDS term structure
well. However, a model with downside risk performs better in capturing conditional stylized
facts because it allows for higher risk aversion in states of bad macroeconomic fundamentals.
Finally, we highlight that the high levels of CDS spreads in states of low expected con-
sumption growth and high macroeconomic uncertainty, which on average occur only 2.3% of
the time, implicitly reflect the tail risk embedded in CDS spreads. For instance, the average
1-year BBB CDS spread is 83 basis points, but there is a huge price discrepancy across
states. The “low-high” state for example has a spread of 552 basis points, but spreads for
the other three states are very low, with a maximum of 187 basis points for times of low ex-
pected growth and macroeconomic uncertainty. Conceptually, this result resembles Berndt
and Obreja (2010), who empirically show that a large fraction of European corporate CDS
returns are explained by a factor mimicking economic catastrophe risk.
VI Conclusion
We show that expected growth and consumption volatility in the U.S. contain information to
account for 75% of the variation of the first two principal components in the term structure
of CDS spreads, which we believe to be a level and slope factor. This information is not
accounted for by a battery of financial market variables such as the VIX index, the VRP,
the excess return on the U.S. stock market, the price-earnings ratio or the investment-grade
and high-yield bond spreads. While some of these variables have individually explanatory
power for the level factor, none of them can account for the variation in the slope factor.
36
To rationalize these empirical findings, we show that a simple equilibrium model with
only two state variables can account for many stylized facts of the sovereign CDS market.
We essentially embed a reduced-form credit risk model into a recursive utility framework.
Time-varying expected growth and macroeconomic uncertainty drive variation in both the
pricing kernel and the default process. Countries differ cross-sectionally by their sensitivities
to aggregate risk. Our model yields tractable closed-form solutions. Countercyclical risk
aversion and a persistent time-varying default process are necessary to match unconditional
moments up to the fourth order and persistence of the CDS term structure, as well as cumu-
lative historical default probabilities at aggregate levels. We extend the model to preferences
with generalized disappointment aversion and show that downside risk improves the fit of
observed downward sloping term structures in states of bad macroeconomic fundamentals.
To the best of our knowledge, the model is the first application of the recursive utility frame-
work to CDS spreads. We also exploit the high-frequency information in the sovereign CDS
market to estimate the preference parameters of the model. The evidence is consistent with
preference for early resolution of uncertainty and a value for the EIS above 1.
Our results emphasize global macroeconomic risk channels as a source of common vari-
ation in the levels of sovereign spreads, beyond the well-documented financial risk channel.
It will be useful to pursue this route to understand the dynamics of the term structure in
relation to both global and local risk factors, such as in Augustin (2012).
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Figure 1: Principal Component Analysis - Factor Loadings
We use principal component analysis to extract the common common factors from the CDS term structure of the 38 countries
in our sample from May 2003 until August 2010. Each bar in the upper and lower panel represents the equally-weighted average
factor loading across the 38 countries in the sample for each CDS contract maturity. The upper panel refers to the loadings on
the first principal component. The lower panel refers to the loadings on the second principal component. Source: Markit.
1 2 3 5 7 100
0.01
0.02
0.03
0.04
0.05Principal Component Analysis - Equally-Weighted Average Factor Loadings
PC1
1 2 3 5 7 10-0.01
0
0.01
0.02
0.03
CDS Contract Maturity
PC2
43
Figure 2: Expected Growth, Consumption Volatility and 5-year Mean CDS Spread
Graph (2a) plots the entire estimated series of monthly filtered conditional expected consumption growth from 1959 to 2010
and figure (2b) traces the entire estimated series of conditional expected consumption volatility. Both series are annualized for
illustration. Grey shaded areas indicate NBER recessions. Graph (2c) plots the historical mean 5-year CDS spread (left scale
- dotted line) of the 38 countries in the sample over the time period May 9th, 2003 until August 19th, 2010 against the filtered
time series of the conditional expected consumption growth (right scale - dashed line) at a monthly horizon. Graph (2d) does
the same for consumption volatility. Data for real per capita consumption is taken from the FRED database of the Federal
Reserve Bank of St.Louis from January 1959 until August 2010. The consumption series are estimated with a traditional
Kalman Filter as described in equation (1). The CDS data are obtained from Markit.
(a) (b)
(c) (d)
44
Figure 3: Model-Implied Conditional CDS Moments
Figure 3 plots the conditional and unconditional model-implied term structure of CDS Spreads for maturities 1 to 10 at the
aggregated level for the rating categories AAA to B. The recovery rate is constant and exogenously set at 25%. The preference
parameters are specified for an investor with a Kreps-Porteus (KP) certainty equivalent. The conditional states are defined
by the combinations of low (µL, σL) and high (µH , σH) expected consumption growth and volatility. The dots reflect the
unconditional moments. We also report the model-implied difference between the 10-year and 1-year CDS spreads (the Slope)
in the worst economic state µLσH . We report the model-implied results for the KP and GDA scenarios, where GDA refers to
an investor with generalized disappointment averse preferences.
Slope = CDS10y − CDS1y: State µLσH (bps)Model AAA AA A BBB BB BKP -9 -14 -26 -45 -92 -93GDA -17 -26 -47 -83 -179 -169
2 4 6 8 100
25
50
75
100
125
Spread (bp)
AAA
2 4 6 8 100
50
100
150
200AA
2 4 6 8 100
50
100
150
200
250
300
Spread (bp)
A
2 4 6 8 100
100
200
300
400
500
600BBB
2 4 6 8 100
200
400
600
800
1000
Spread (bp)
Maturity
BB
2 4 6 8 10
300
600
900
1200
1500
B
Spread (bp)
Maturity
µµµµLσσσσL
µµµµLσσσσH
µµµµHσσσσL
µµµµHσσσσH Mean
45
Figure 4: Reversal of the CDS Term Structure
Figure 4 plots the conditional and unconditional model-implied term structure of CDS Spreads for maturities 1 to 10 at the
aggregated level for the rating categories AA, A and B, as well as the historical difference between the 10 year and 1 year CDS
spread (Slope of the term structure) for Portugal, Italy, Greece and Spain. The recovery rate is constant and exogenously set
at 25%. The preference parameters are specified for an investor with a Kreps Porteus certainty equivalent. The conditional
states are defined by the combinations of low (µL, σL) and high (µH , σH) expected consumption growth and volatility. The
dots reflect the unconditional mean.
2 4 6 8 100
50
100
150
200Conditional Term Structure AA
Spread (bp)
Maturity05/2003 02/2004 11/2004 08/2005
-500
-400
-300
-200
-100
0
100
Time
Spread (bp)
Slope
Portugal
Italy
Greece
Spain
2 4 6 8 100
50
100
150
200
250
300
Spread (bp)
Maturity
Conditional Term Structure A
2 4 6 8 100
200
400
600
800
1000
1200
1400
1600
Conditional Term Structure B
Spread (bp)
Maturity
µµµµLσσσσL
µµµµLσσσσH
µµµµHσσσσL
µµµµHσσσσH Mean
46
Figure 5: Hazard Rate Analysis
Figure 5 plots the unconditional observed and model-implied term structure of CDS Spreads for maturities 1 to 10 at the
aggregated level for the rating categories AAA-B for various specifications of the hazard rate ht = λt/ (1 + λt), where λt =
exp (βλ0 + βλxxt + βλσσt). The dash-dotted line represents the results using the benchmark specification of the hazard rate,
the dashed line denotes the results using a constant hazard rate, whereas the dotted and solid lines feature the results for a
default process specification with only expected consumption growth or macroeconomic uncertainty respectively. The empirical
observations in the data are represented with the solid bullet points. The recovery rate is constant and exogenously set at 25%.
The preference parameters are specified for an investor with Kreps Porteus preferences.
2 4 6 8 1010
15
20
25
30
35
Spread (bp)
AAA
2 4 6 8 1020
30
40
50
60AA
2 4 6 8 1030
40
50
60
70
80
Spread (bp)
A
2 4 6 8 1070
90
110
130
150
170
BBB
2 4 6 8 10100
150
200
250
300
Spread (bp)
Maturity
BB
2 4 6 8 10400
450
500
550
600
650B
Spread (bp)
Maturity
lnλλλλt=ββββ
λλλλ 0+ββββ
λλλλ σσσσ σσσσ
tlnλλλλ
t=ββββ
λλλλ 0+ββββ
λλλλ x x
tlnλλλλ
t=ββββ
λλλλ 0+ββββ
λλλλ x x
t+ββββ
λλλλ σσσσ σσσσ
tlnλλλλ
t=ββββ
λλλλ 0 Data
47
Table 1: Country List
Table 1 presents the list of 38 countries in the sample and the corresponding geographical region. We map a country’s rating
into a numerical scheme ranging from 1 (AAA) to 21 (C). For the analysis, countries are grouped into 6 major rating buckets
(AAA, AA, A, BBB, BB, B), keeping track of rating changes over time. The third column indicates the Standard&Poor’s
Rating in 2010. The fourth column indicates the average historical rating as traced back by Fitch Ratings over the sample
period, that is from May 9th, 2003 until August 19th, 2010. The equally weighted historical average of the numerical rating is
rounded to the nearest integer to obtain the average final rating. The last two columns indicate the average number of countries
in each rating group.
Country Region S&P Rating (’10) Average Rating Classification Average # entitiesAustria Europe AAA AAA
AAA 4France Europe AAA AAAGermany Europe AAA AAASpain Europe AA AAABelgium Europe AA+ AA+
AA 6
Italy Europe A+ AA-Japan Asia AA AAPortugal Europe A- AAQatar Middle East AA AASlovenia E.Europe AA AA-Chile Lat.Amer A+ A-
A 9
China Asia A+ ACzech Republic E.Europe A AGreece Europe BB+ AIsrael Middle East A A-Korea (Republic of) Asia A A+Lithuania E.Europe BBB A-Malaysia Asia A- A-Slovakia E.Eur A+ ABulgaria E.Eur BBB BBB-
BBB 11
Croatia E.Europe BBB BBB-Hungary E.Europe BBB- BBB+Mexico Lat.Amer BBB BBB+Morocco Africa BBB- BBB-Panama Lat.Amer BBB- BBB-Poland E.Europe A- BBB+Romania E.Europe BB+ BBB-Russian Federation E.Europe BBB BBB+South Africa Africa BBB+ BBB+Thailand Asia BBB+ BBB+Brazil Lat.Amer BBB- BB
BB 6
Colombia Lat.Amer BB+ BBEgypt Africa BB+ BB+Peru Lat.Amer BBB- BB+Philippines Asia BB- BBTurkey Middle East BB BB-Lebanon Middle East B B-
B 2Venezuela Lat.Amer BB- B+
48
Tab
le2:
Sum
mar
ySta
tist
ics
Tab
le2
rep
ort
ssu
mm
ary
stati
stic
sfo
rth
eC
DS
term
stru
ctu
reof
38
sover
eign
cou
ntr
ies
over
the
sam
ple
per
iod
May
9,
2003
unti
lA
ugu
st19,
2010.
All
CD
Ssp
read
sare
mid
com
posi
tequ
ote
san
dU
SD
den
om
inate
d.
Cou
ntr
ies
are
aggre
gate
db
ase
don
6m
ajo
rra
tin
gca
tegori
esra
ngin
gfr
om
AA
Ato
B.
At
each
date
,all
ob
serv
ati
on
sw
ith
ina
giv
en
rati
ng
cate
gory
are
aggre
gate
dby
takin
gth
eeq
ually-w
eighte
daver
age
CD
Ssp
read
.F
or
the
6aggre
gate
dsp
read
seri
es,
the
tab
lere
port
sth
em
ean
,m
edia
n,
stan
dard
dev
iati
on
,
min
imu
m,
maxim
um
and
the
firs
t-ord
erau
toco
rrel
ati
on
coeffi
cien
t(A
C1).
Sou
rce:
Mark
it.
AAA
1y
2y
3y
5y
7y
10y
AA
1y
2y
3y
5y
7y
10y
Mean
14
16
18
22
23
25
24
28
31
38
41
45
Median
22
34
57
57
10
14
19
24
Sta
nd.d
ev.
23
25
27
31
31
31
38
40
42
45
45
44
Min
imum
01
12
23
12
35
68
Maxim
um
128
135
140
153
153
152
170
183
194
207
209
206
AC1
0.9
930
0.9
944
0.9
960
0.9
970
0.9
970
0.9
971
0.9
956
0.9
961
0.9
965
0.9
968
0.9
968
0.9
967
A1y
2y
3y
5y
7y
10y
BBB
1y
2y
3y
5y
7y
10y
Mean
35
42
48
60
65
71
79
95
108
130
141
152
Median
14
19
24
34
41
49
29
47
66
93
110
127
Sta
nd.d
ev.
51
55
58
63
62
61
106
108
107
104
101
97
Min
imum
46
710
13
17
11
15
20
30
37
46
Maxim
um
281
294
325
351
362
370
540
568
586
608
617
628
AC1
0.9
973
0.9
974
0.9
974
0.9
978
0.9
976
0.9
976
0.9
973
0.9
973
0.9
971
0.9
969
0.9
969
0.9
967
BB
1y
2y
3y
5y
7y
10y
B1y
2y
3y
5y
7y
10y
Mean
129
168
202
255
281
303
416
472
510
558
572
593
Median
91
143
182
244
274
294
346
399
440
489
516
540
Sta
nd.d
ev.
141
138
133
127
122
118
320
312
302
287
265
248
Min
imum
27
44
60
97
122
146
75
135
165
202
234
275
Maxim
um
1056
1077
1075
1063
1061
1065
2039
1984
1941
1881
1828
1800
AC1
0.9
954
0.9
951
0.9
948
0.9
943
0.9
939
0.9
937
0.9
931
0.9
933
0.9
933
0.9
934
0.9
930
0.9
924
49
Table 3: Principal Component Analysis
Table 3 reports the variation in CDS spreads explained by the first 6 factors PC1 to PC6 of the principal component analysis.
The row All refers to the pooled data, where all six maturities for all 38 countries are taken together. Subsequent columns
indicate results for the subsamples, taken by contract maturity (1y to 10y) each at a time. Source: Markit.
PC1 PC2 PC3 PC4 PC5 PC6All 77.8158 91.0749 94.7448 96.3491 97.5028 98.23781y 85.9812 92.8245 95.7170 97.1810 98.2461 99.05402y 83.0337 91.6612 95.5640 97.1889 98.2693 98.92293y 79.7215 91.7345 95.5324 97.1032 98.1849 98.82075y 75.1912 92.0572 95.2786 96.7295 97.9724 98.70117y 72.8903 91.5746 94.8203 96.2861 97.4767 98.577910y 70.4393 91.6796 94.6215 96.2402 97.5720 98.4832
50
Table 4: Kalman Filter Estimates
Table 4 reports the Kalman Filter estimates for the parameters of the conditional expectation of consumption growth and
conditional consumption volatility for the system of equations
gt+1 = xt + σtεg,t+1
xt+1 = (1− φx)µx + φxxt + νxσtεx,t+1
σ2t+1 = (1− φσ)µσ + φσσ
2t + νσεσ,t+1,
where gt, xt and σt reflect the dynamics for, respectively, aggregate consumption growth, its conditional mean and volatility,
and all error terms are standard normal. In addition to the short-run consumption shocks εg,t+1, consumption growth is fed
with long-run shocks εx,t+1, whose persistence is defined through the parameter φx. The unconditional mean growth-rate
is given by µx and νx denotes the sensitivity of expected growth to long-run shocks. The parameters φσ and νσ define the
persistence of and sensitivity to shocks εσ,t+1 to macroeconomic uncertainty, which is fluctuating around its long-run mean µσ .
Macroeconomic uncertainty is defined as a GARCH-like stochastic volatility process that has been used in Heston and Nandi
(2000), henceforth HN. Panel A reports the results for this base specification. Standard errors are given in parentheses. Panel
B reports the estimated coefficients and standard errors in parentheses for alternative volatility specifications: the GARCH(1,1)
model of Bollerslev (1986), the EGARCH model of Nelson (1991) and the GJR GARCH model of Glosten, Jagannathan, and
Runkle (1993), which have the following dynamics:
GARCH(1, 1) : σ2t+1 = (1− φσ)µσ + φσσ
2t + νσσ
2t
(ε2c,t+1 − 1
)EGARCH : lnσ2
t+1 = (1− φσ) lnµσ + φσ lnσ2t + νσ
(|εc,t+1| −
√2/π
)+ λσεc,t+1
GJR : σ2t+1 = (1− φσ)µσ + φσσ
2t + νσσ
2t
(ε2c,t+1 − 1
)+ λσσ
2t
(ε2c,t+1I (εc,t+1 < 0)−
1
2
),
where the additional leverage parameter λσ in the EGARCH and GJR-GARCH specifications allows for asymmetric effects of
positive and negative shocks to volatility. All coefficients are estimated using data for real per capita consumption growth from
the FRED database of the Federal Reserve Bank of St.Louis from January 1959 until August 2010.
Panel A µx φx νx µσ φσ νσ
HN0.001785 0.955642 0.058611 1.372177e-05 0.9610790 7.5528e-007
(0.000235) (0.033936) (0.028885) (1.541653e-06) (0.013410) (1.7537e-007)
Panel B µσ φσ νσ λσ
GARCH(1,1)9.0536e-06 0.97791 0.03598 –
(1.0199e-05) (0.00364) (0.00840) (–)
EGARCH1.5787e-05 0.99068 0.09589 0.02269
(4.5563e-06) (0.00460 (0.02338) (0.00969)
GJR1.3293e-05 0.98495 0.07086 −0.03128
(1.6063e-05) (0.00554) (0.01731) (0.01736)
51
Table 5: Regression Analysis - Macroeconomic and Financial Risk
Table 5 reports the results from the regression of the monthly averages of the factors extracted from a principal component analysis onto conditional expected consumption
growth, conditional consumption volatility and the Variance Risk Premium (VRP), the CBOE S&P500 volatility index (VIX), the excess return on the CRSP value-weighted
portfolio (USret), the US price-earnings ratio (PE), as well as the U.S. investment-grade (AAA BBB) and high-yield (BBB BB) bond spreads. Data for real per capita
consumption is from January 1959 until August 2010 from the FRED database of the Federal Reserve Bank of St.Louis. The VRP is from Hao Zhou’s webpage, the USret from
Kenneth French’s website, the PE from Robert Shiller’s website, and the VIX, AAA BBB and BBB BB from the FRED H15 report.
Fi,t = a0,i + a1,i × xt|t + a2,i × σt + a3,i × V RPt + a4,i × V IXt + a5,i × USrett + a6,i × PEt + a7,i ×AAA BBBt + a8,i ×BBB BBt + εt,
where i = 1, 2 and t is the month index. The dependent variables Fi,t denote the principal components, xt|t is the filtered expected consumption growth and σt the filtered
conditional consumption volatility. The last row reports the decision rule (A = Accept, R = Reject) from a Dickey-Fuller test on the residuals to test for the presence of a
unit root. Columns (15) to (17) report the results of the probit model that tests whether the explanatory variables can predict the Reinhart and Rogoff crisis tally indicator,
which is available for 29 countries in our sample. The crisis tally is a count indicator accounting for currency crises, inflation crises, stock market crashes, domestic and external
sovereign debt crises, and banking crises. More specifically, we fit the regression specification.
Pr (Crisis = 1) = Φ(a0 + a1xt|t + a2σt + a3V RPt + a4V IXt + a5USrett + a6PEt + a7BBB AAAt + a8BB BBBt
).
Block-bootstrapped standard errors are reported in brackets. ∗∗ and ∗ indicate significance at the 1% and 5% respectively.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17)VARIABLES F1 F2 F1 F2 F1 F2 F1 F2 F1 F2 F1 F2 F1 F2 CI CI CI
xt|t -128.16** 236.66** -0.22** -0.22** -0.22
(38.92) (22.10) (0.06) (0.07) (0.15)σt 454.81** 293.54** 0.45** 0.42* 0.99**
(77.58) (34.70) (0.17) (0.18) (0.37)VRP 23.68 -2.28 -0.01
(22.79) (5.25) (0.00)VIX 1.44** -0.15 0.04**
(0.37) (0.11) (0.01)USret -0.46 0.07 -0.01** -0.02
(0.91) (0.20) (0.00) (0.01)PE -0.04** 0.01* -0.00 0.04**
(0.00) (0.00) (0.00) (0.01)AAA BBB 17.80** -1.80 0.04
(1.71) (1.14) (0.14)BBB BB 16.84** -1.45 -0.11
(2.05) (1.18) (0.13)Constant -1.28** -0.77** -0.04 0.00 -0.30** 0.03 0.00 -0.00 1.03** -0.16* -0.25** 0.03 -0.30** 0.03 -0.59 -0.50 -3.58**
(0.20) (0.09) (0.04) (0.01) (0.06) (0.03) (0.03) (0.01) (0.12) (0.07) (0.02) (0.02) (0.03) (0.02) (0.42) (0.48) (0.76)
Observations 88 88 88 88 88 88 88 88 88 88 88 88 88 88 18,096 18,096 11,830R2 0.76 0.74 0.13 0.01 0.56 0.04 0.01 0.00 0.78 0.10 0.83 0.05 0.76 0.03adj.R2 0.75 0.74 0.12 -0.00 0.55 0.02 0.00 -0.01 0.78 0.09 0.82 0.04 0.76 0.02ps.R2 0.01 0.01 0.07Dickey-Fuller R** R** R A R** A A A R A R** A R** A
52
Table 6: Regression Analysis - Macroeconomic and Financial Risk
Table 6 reports the results from the regression of the factors extracted from a principal component analysis onto conditional expected consumption growth, conditional
consumption volatility and the Variance Risk Premium (VRP), the CBOE S&P500 volatility index (VIX), the excess return on the CRSP value-weighted portfolio (USret), the
US price-earnings ratio (PE), as well as the U.S. investment-grade (AAA BBB) and high-yield (BBB BB) bond spreads. Factor scores are first averaged at the end of each
month and then projected onto the explanatory variables. Data for real per capita consumption is taken from the FRED database of the Federal Reserve Bank of St.Louis from
January 1959 until August 2010. The data for the VRP is taken from Hao Zhou’s webpage, for the USret on Kenneth French’s website, the PE from Robert Shiller’s website,
and the VIX, AAA BBB and BBB BB from the FRED H15 report. Block-bootstrapped standard errors are reported in brackets. ∗∗ and ∗ indicate significance at the 1% and
5% respectively.
Fi,t = a0,i + a1,i × xt|t + a2,i × σt + a3,i × V RPt + a4,i × V IXt + a5,i × USrett + a6,i × PEt + a7,i ×AAA BBBt + a8,i ×BBB BBt + εt,
where i = 1, 2, 3 and t is the month index. The dependent variables Fi,t denote the principal components, xt|t is the filtered expected consumption growth and σt the filtered
conditional consumption volatility. The last row reports the decision rule (A = Accept, R = Reject) from a Dickey-Fuller test on the residuals to test for the presence of a unit
root.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)VARIABLES F1 F2 F1 F2 F1 F2 F1 F2 F1 F2 F1 F2 F1 F2
xt|t -120.99* 239.89** -69.26 248.26** -118.87** 238.30** 164.96* 260.42** 70.69 255.43** 29.07 266.36** 136.96* 247.91**(47.79) (23.65) (62.20) (22.31) (39.96) (22.39) (69.97) (24.13) (45.69) (20.23) (40.92) (25.27) (67.02) (29.92)
σt 450.18** 291.46** 401.68** 283.08** 469.60** 296.15** 286.17** 279.87** 254.52** 274.64** 348.96** 273.55** 255.53** 280.00**(81.73) (36.66) (60.58) (38.47) (82.80) (34.99) (53.98) (39.20) (59.90) (43.13) (52.07) (33.28) (56.98) (47.37)
VRP 3.64 1.63 0.98 3.38(13.18) (1.82) (9.57) (4.79)
VIX 0.48 0.09 0.06 -0.05(0.40) (0.07) (0.33) (0.12)
USret -0.55 -0.10 0.11 0.20(0.36) (0.10) (0.33) (0.20)
PE -0.04** -0.00 -0.02 0.00(0.01) (0.00) (0.01) (0.01)
AAA BBB 14.75** 1.39 5.56 -0.76(2.73) (1.11) (4.42) (2.49)
BBB BB 11.25** 2.12* 5.94* 4.06(1.98) (0.98) (2.90) (2.28)
Constant -1.27** -0.76** -1.22** -0.76** -1.31** -0.77** 0.29 -0.64** -0.89** -0.73** -1.15** -0.74** -0.52 -0.90**(0.21) (0.09) (0.17) (0.10) (0.22) (0.09) (0.27) (0.16) (0.15) (0.11) (0.14) (0.09) (0.33) (0.22)
Observations 88 88 88 88 88 88 88 88 88 88 88 88 88 88R2 0.76 0.75 0.78 0.75 0.77 0.75 0.87 0.75 0.88 0.75 0.88 0.77 0.91 0.78adj.R2 0.75 0.74 0.77 0.74 0.77 0.74 0.86 0.74 0.87 0.74 0.88 0.76 0.90 0.76Dickey-Fuller R** R** R** R** R** R** R** R* R** R R** R* R** R*
53
Table 7: Country Regressions - Macroeconomic Risk and Variance Risk Premium
Table 7 reports the results from the regression of the level and slope of the monthly sovereign CDS series onto conditional expected consumption growth, conditional consumption
volatility and the Variance Risk Premium (VRP). The level is defined as the average monthly CDS spread over all maturities, the slope is equal to the difference between the
average monthly 10-year and 1-year CDS spread. Data for real per capita consumption is taken from the FRED database of the Federal Reserve Bank of St.Louis from January
1959 until August 2010. The consumption series are estimated using a traditional Kalman Filter as described in equation (1). The data for the VRP are taken from Hao Zhou’s
webpage. We specify the regression model
Yi,t = a0,i + a1,i × xt|t + a2,i × σt + a3,i × V RPt + εt
where Yi,t is either the Level or the Slope of the CDS curve, i denotes the country and t is the month index. xt|t is the filtered expected consumption growth, σt the filtered
conditional consumption volatility and V RPt denotes the VRP. The average correlation between the level and slope across countries is 0.12. All regressions are run for each
of the 38 countries. Panel A excludes the VRP, Panel B excludes the consumption predictors, and Panel C includes all explanatory variables. Columns (1) to (3) report the
fraction (out of 38) of 5% statistically significant coefficient estimates, while columns (4) to (6) report the fraction (out of 38) of positive coefficient estimates. Columns (7) to
(8) report respectively the mean and median adjusted R2 of the 38 country-pecific regressions. Column (9) indicates the number of observations for each regression. Columns
(10) to (12) report the signs of a1 and a2, as well as the median R2 from the population regressions predicted by the model.
Data ModelPanel A t-stats xt|t t-stats σt t-stats V RPt +xt|t +σt,t +V RPt mean adj.R2 median adj.R2 # obs. Sign a1 Sign a2 median R2
Level 0.82 0.89 0.32 0.97 0.69 0.70 88 - + 0.97Slope 0.84 0.61 0.55 0.74 0.46 0.54 88 + + 0.75
Panel BLevel 0.95 1.00 0.11 0.12 88Slope 0.50 0.69 0.04 0.03 88
Panel CLevel 0.76 0.89 0.18 0.32 0.97 1.00 0.69 0.70 88Slope 0.84 0.58 0.11 0.61 0.74 0.84 0.47 0.54 88
Regression (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
54
Table 8: Parameters of the Markov-Switching Model
Consumption growth dynamics are calibrated at a monthly frequency as in Bansal, Kiku, and Yaron (2012) with µx = 0.0015,
φx = 0.975, νx = 0.038,√µσ = 0.0072, φσ = 0.999 and νσ = 2.80 × 10−6. Conditional on the information set Jt, the errors
are i.i.d. normal.
gt+1 = xt + σtεg,t+1
xt+1 = (1− φx)µx + φxxt + νxσtεx,t+1
σ2t+1 = (1− φσ)µσ + φσσ2
t + νσεσ,t+1
εg,t+1
εx,t+1
εσ,t+1
| Jt ∼ NID
0
0
0
,
1 0 0
0 1 0
0 0 1
.
To be consistent with the daily frequency of CDS spreads, we map the monthly parameters into daily values such that annual
moments of consumption growth are preserved. We use the monthly-to-daily mapping
µdailyx = ∆µx, φdailyx = φ∆x , νdailyx = νx
√√√√√(1− φ2∆x
1− φ2x
)/1 +2φx
1− φx−
2∆φx(
1− φ1/∆x
)(1− φx)2
µdailyσ = ∆µσ , φdailyσ = φ∆σ , νdailyσ = νσ
√∆
√√√√√(1− φ2∆σ
1− φ2σ
)/1 +2φσ
1− φσ−
2∆φσ(
1− φ1/∆σ
)(1− φσ)2
where we consider ∆ = 1/22, i.e. 22 trading days per month. The parameters at a daily frequency obtained from the
mapping system are µdailyx = 6.8182 × 10−5, φdailyx = 0.9988, νdailyx = 0.0019, µdailyσ = 2.3564 × 10−6, φdailyσ = 0.99995
and νdailyσ = 2.7247× 10−8. Panel A reports the parameters of the four-state daily Markov-switching model. The conditional
mean and variance of consumption growth are µg and ωg respectively. P> is the transition matrix across different regimes and
Π is the vector of unconditional probabilities of regimes. The four states are characterized by the combinations of expected
consumption growth (µ) and consumption volatility (σ), which can be high (H) and low (L). Panel B and C report the (time-
averaged) annualized model statistics for aggregate consumption growth, the risk premium and the volatility of the aggregate
stock market, the risk-free rate and its volatility, all compared against observed values. The model-implied values are reported
for both the model with the Kreps-Porteus certainty equivalent (KP) and generalized disappointment averse preferences (GDA).
The estimated values are sampled on an annual frequency from 1930 to 2008. Standard errors are reported in parentheses.
Panel A µLσL µLσH µHσL µHσHµ>g -0.00011 -0.00011 0.00009 0.00009(
ω>g)1/2
0.00094 0.00281 0.00094 0.00281
P>
µLσL 0.99897 0.00001 0.00102 0.00000µLσH 0.00004 0.99894 0.00000 0.00102µHσL 0.00013 0.00000 0.99986 0.00001µHσH 0.00000 0.00013 0.00004 0.99984
Π> 0.08600 0.02304 0.70268 0.18828Panel B Model DataE[∆g] 1.80 1.92 (0.33)σ[∆g] 2.53 2.12 (0.52)
AC1[∆g] 0.46 0.46 (0.15)Panel C Model (KP) Model (GDA) Data
E[rm − rf ] + 0.5σ2[rm] 6.38 5.52 7.84 (1.97)σ[rm] 18.48 17.48 20.16 (2.14)E[rf ] 1.22 1.01 0.86 (0.89)σ[rf ] 1.05 0.80 1.74 (0.33)
55
Table 9: Model Estimation - Preference and Default Parameters
Table 9 reports estimation results (with with Newey-West standard errors in parentheses) for preference parameters and the
parameters of the default process for the rating categories AAA to B, where the default process ht is defined as ht = λt/ (1 + λt),
where λt = exp (βλ0 + βλxxt + βλσσt). The subjective discount factor δ is fixed at 0.9989. γ is the coefficient of relative risk
aversion, ψ the elasticity of intertemporal substitution, α the disappointment aversion parameter and κ denotes the fraction of
the certainty equivalent below which outcomes are disappointing. The estimation is carried out via the Generalized Method of
Moments using the historical observed time series of credit default swap spreads over the sample period 9 May 2003 through 19
August 2010. We do the estimation using both spread levels and changes. The moments in the estimation are the expectations
of the CDS spreads (respectively spread changes) and their squared values. The weighting matrix is the inverse of the diagonal of
the spectral density matrix. The last panel reports the J statistic for the test of overidentifying restrictions and the corresponding
p-value, as well as the likelihood-ratio test (LR) where the null hypothesis is that the true preferences are Kreps-Porteus against
the alternative that they are generalized disappointment averse.
Kreps-Porteus (KP)AAA AA A BBB BB B
βλ0−15.37 −13.71 −12.71 −11.34 −10.07 −9.15
(s.e.) (0.66) (0.24) (0.14) (0.09) (0.09) (0.06)
βλx −5, 624.18 −5, 596.99 −7, 429.56 −6, 692.67 −13, 917.70 −4, 144.73(s.e.) (415.62) (514.33) (680.13) (730.82) (1, 928.74) (415.18)
βλσ 1, 818.66 1, 390.45 1, 125.74 886.99 309.57 583.87(s.e.) (228.44) (80.70) (68.55) (46.44) (131.77) (40.24)
Generalized Disappointment Aversion (GDA)AAA AA A BBB BB B
βλ0−15.44 −13.79 −12.89 −11.48 −10.90 −9.21
(s.e.) (0.58) (0.24) (0.15) (0.10) (0.41) (0.05)
βλx −6, 572.55 −6, 466.87 −8, 686.44 −7, 669.86 −18, 598.21 −4, 431.00(s.e.) (467.94) (663.93) (850.56) (902.24) (4, 858.26) (433.62)
βλσ 1, 812.21 1, 388.35 1, 143.98 901.76 467.94 593.74(s.e.) (198.52) (75.36) (64.03) (42.32) (73.04) (37.71)
Preference ParametersLevels Differences
Parameter KP GDA KP GDAδ 0.9989 0.9989 0.9989 0.9989
(s.e.) ( - ) ( - ) ( - ) ( - )γ 8.2692 4.9081 7.1713 5.0614
(s.e.) ( 0.0011 ) ( 0.0054 ) ( 0.0642 ) ( 0.1294 )ψ 1.5774 1.4874 1.5953 1.2473
(s.e.) ( 0.0002 ) ( 0.0018 ) ( 0.0133 ) ( 0.0273 )α 1.0000 0.2486 1.0000 0.4819
(s.e.) ( - ) ( 0.0008 ) ( - ) (0.0107)κ - 0.8470 - 0.9549
(s.e.) (- ) ( 0.0042 ) ( - ) ( 0.0183 )Model Statistics
J − test 45.73 14.65 19.84 8.32p− value 0.65 1.00 1.00 1.00LR− test 31.08 11.52p− value <0.01 <0.01
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Table 10: Model-Implied and Observed Term Structure of CDS Spreads AAA-B
Table 10 reports observed and model-implied unconditional means, standard deviations (in basis points), skewness, kurtosis and first-order autocorrelation coefficients for CDS
spreads for maturities 1 to 10 for the rating categories AAA to B. The column labeled RMSE reports the root mean squared errors in basis points for the model fit. The
recovery rate has a constant value of 25%. Preference parameters are those of an investor with a Kreps-Porteus certainty equivalent.
MODEL DATA
AAA 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 14 16 17 20 23 27 1.06 14 16 18 22 23 25Volatility 25 26 27 29 30 31 1.50 23 25 27 31 31 31Skewness 2 2 2 2 2 2 0.15 2 2 2 2 2 2Kurtosis 9 7 6 5 4 3 1.12 7 7 6 5 5 4AC1 0.9997 0.9998 0.9998 0.9999 0.9999 0.9999 0.0044 0.9930 0.9944 0.9960 0.9970 0.9970 0.9971
AA 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 25 28 31 36 40 46 0.97 24 28 31 38 41 45Volatility 38 39 41 43 45 47 1.28 38 40 42 45 45 44Skewness 2 2 2 2 2 2 0.21 2 2 2 2 2 2Kurtosis 10 8 6 5 4 4 1.74 6 6 5 5 4 4AC1 0.9997 0.9998 0.9998 0.9999 0.9999 0.9999 0.0034 0.9956 0.9961 0.9965 0.9968 0.9968 0.9967
A 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 37 42 47 56 64 73 1.89 35 42 48 60 65 71Volatility 52 54 56 59 62 64 2.14 51 55 58 63 62 61Skewness 3 3 2 2 2 2 0.35 3 2 2 2 2 2Kurtosis 17 13 10 7 5 4 3.41 10 9 8 7 7 7AC1 0.9995 0.9996 0.9997 0.9998 0.9999 0.9999 0.0022 0.9973 0.9974 0.9974 0.9978 0.9976 0.9976
BBB 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 86 97 107 124 138 154 4.11 79 95 108 130 141 152Volatility 95 98 101 105 108 111 9.33 106 108 107 104 101 97Skewness 3 3 2 2 2 2 0.48 2 2 2 2 2 2Kurtosis 15 12 9 7 5 4 4.04 7 7 7 6 6 7AC1 0.9995 0.9996 0.9997 0.9998 0.9998 0.9999 0.0027 0.9973 0.9973 0.9971 0.9969 0.9969 0.9967
BB 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 136 170 199 244 278 314 7.23 129 168 202 255 281 303Volatility 159 146 136 121 111 100 11.89 141 138 133 127 122 118Skewness 3 3 3 3 3 2 0.33 3 3 3 2 2 2Kurtosis 12 12 12 12 11 10 1.85 15 14 13 11 11 10AC1 0.9989 0.9989 0.9990 0.9991 0.9992 0.9993 0.0046 0.9954 0.9951 0.9948 0.9943 0.9939 0.9937
B 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 442 473 498 539 569 600 14.34 416 472 510 558 572 593Volatility 282 285 287 289 289 287 27.20 320 312 302 287 265 248Skewness 2 2 2 2 2 2 0.35 2 2 2 2 2 2Kurtosis 8 7 6 5 5 4 3.33 10 9 9 8 9 9AC1 0.9997 0.9997 0.9998 0.9998 0.9998 0.9999 0.0067 0.9931 0.9933 0.9933 0.9934 0.9930 0.9924
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Table 11: Model-Implied Term Structure of Default Probabilities AAA-B
Table 11 reports unconditional model-implied physical and risk-neutral default probabilities for maturities 1 to 10 at the
aggregated level for the rating categories AAA to B as well as their ratio. The column labeled RMSE reports the root mean
squared errors in % points for the model fit. Model results are compared against the observed Standard&Poor’s sovereign
foreign-currency cumulative average default rates over the time frame 1975 to 2009 for the physical default probabilities. The
recovery rate has a constant value of 25%.
RMSE 1y 2y 3y 4y 5y 6y 7y 8y 9y 10yPhysical Default Probabilities
AAAModel 0.97 0.16 0.32 0.48 0.63 0.79 0.94 1.10 1.25 1.40 1.55Observed − 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
AAModel 1.73 0.29 0.57 0.86 1.14 1.41 1.69 1.96 2.23 2.50 2.77Observed − 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
AModel 2.40 0.40 0.80 1.19 1.58 1.96 2.34 2.72 3.09 3.46 3.83Observed − 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
BBBModel 1.37 0.96 1.90 2.82 3.73 4.62 5.49 6.34 7.19 8.01 8.83Observed − 0.00 0.50 1.57 2.72 3.97 5.33 6.07 6.07 6.07 6.07
BBModel 2.02 1.25 2.45 3.61 4.74 5.84 6.92 7.99 9.03 10.06 11.08Observed − 0.74 2.36 3.70 4.70 6.36 8.24 10.34 12.72 13.63 13.63
BModel 11.11 5.19 10.00 14.49 18.68 22.60 26.28 29.75 33.00 36.08 38.98Observed − 2.13 5.03 6.71 9.32 11.67 13.54 15.85 20.06 21.87 24.57
Risk-Neutral Default ProbabilitiesAAA Model − 0.18 0.41 0.68 0.98 1.32 1.68 2.06 2.47 2.89 3.34AA Model − 0.33 0.74 1.21 1.73 2.30 2.91 3.56 4.24 4.94 5.66A Model − 0.48 1.11 1.84 2.68 3.59 4.56 5.58 6.65 7.75 8.87BBB Model − 1.13 2.52 4.10 5.84 7.69 9.62 11.60 13.62 15.65 17.69BB Model − 1.78 4.35 7.46 10.95 14.67 18.50 22.39 26.25 30.05 33.76B Model − 5.65 11.59 17.58 23.47 29.15 34.56 39.66 44.44 48.88 53.01
Ratio of Risk-Neutral to Physical Default ProbabilitiesAAA Model − 1.34 1.69 2.03 2.36 2.68 2.98 3.26 3.52 3.76 3.98AA Model − 1.22 1.44 1.65 1.84 2.03 2.21 2.37 2.53 2.67 2.80A Model − 1.27 1.51 1.72 1.92 2.11 2.27 2.43 2.57 2.70 2.82BBB Model − 1.21 1.38 1.54 1.68 1.80 1.91 2.00 2.09 2.16 2.22BB Model − 1.75 2.22 2.53 2.76 2.93 3.06 3.16 3.23 3.28 3.32B Model − 1.09 1.17 1.23 1.28 1.32 1.35 1.37 1.38 1.39 1.40
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Table 12: Model-Implied and Observed Term Structure of CDS Spreads AAA-B: Downside Risk Aversion
Table 12 reports observed and model-implied unconditional means, standard deviations (in basis points), skewness, kurtosis and first-order autocorrelation coefficients for CDS
spreads for maturities 1 to 10 for the rating categories AAA-B. The column labeled RMSE reports the root mean squared errors in basis points for the model fit. The recovery
rate has a constant value of 25%. Preference parameters are those of an investor with Generalized Disappointment Aversion.
MODEL DATA
AAA 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 13 16 18 21 23 26 0.73 14 16 18 22 23 25Volatility 24 26 27 29 30 31 1.13 23 25 27 31 31 31Skewness 3 2 2 2 1 1 0.18 2 2 2 2 2 2Kurtosis 10 7 5 4 3 3 1.49 7 7 6 5 5 4AC1 0.9997 0.9998 0.9999 0.9999 0.9999 0.9999 0.0044 0.9930 0.9944 0.9960 0.9970 0.9970 0.9971
AA 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 24 28 31 37 41 45 0.52 24 28 31 38 41 45Volatility 37 39 41 44 46 46 1.17 38 40 42 45 45 44Skewness 3 2 2 2 2 1 0.21 2 2 2 2 2 2Kurtosis 10 7 5 4 3 3 1.84 6 6 5 5 4 4AC1 0.9997 0.9998 0.9999 0.9999 0.9999 0.9999 0.0034 0.9956 0.9961 0.9965 0.9968 0.9968 0.9967
A 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 35 43 49 58 65 72 0.90 35 42 48 60 65 71Volatility 51 54 56 61 63 64 1.91 51 55 58 63 62 61Skewness 3 3 2 2 2 1 0.42 3 2 2 2 2 2Kurtosis 17 11 7 5 4 3 3.65 10 9 8 7 7 7AC1 0.9995 0.9997 0.9998 0.9999 0.9999 0.9999 0.0023 0.9973 0.9974 0.9974 0.9978 0.9976 0.9976
BBB 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 83 97 109 127 139 152 2.35 79 95 108 130 141 152Volatility 93 97 101 106 109 111 9.99 106 108 107 104 101 97Skewness 3 2 2 2 2 2 0.45 2 2 2 2 2 2Kurtosis 15 10 7 5 4 4 3.89 7 7 7 6 6 7AC1 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 0.0028 0.9973 0.9973 0.9971 0.9969 0.9969 0.9967
BB 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 128 171 204 250 279 307 3.32 129 168 202 255 281 303Volatility 155 140 131 122 118 115 6.39 141 138 133 127 122 118Skewness 4 3 3 2 2 2 0.25 3 3 3 2 2 2Kurtosis 18 17 15 11 8 6 2.31 15 14 13 11 11 10AC1 0.9990 0.9991 0.9993 0.9995 0.9996 0.9997 0.0049 0.9954 0.9951 0.9948 0.9943 0.9939 0.9937
B 1 2 3 5 7 10 RMSE 1 2 3 5 7 10Mean 434 473 503 545 571 596 9.60 416 472 510 558 572 593Volatility 279 284 287 290 289 287 28.31 320 312 302 287 265 248Skewness 2 2 2 2 2 2 0.46 2 2 2 2 2 2Kurtosis 8 6 5 4 4 4 4.16 10 9 9 8 9 9AC1 0.9997 0.9998 0.9998 0.9999 0.9999 0.9999 0.0068 0.9931 0.9933 0.9933 0.9934 0.9930 0.9924
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